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US Army Corps 
of Engineers 

Water Resources Support Center 


RISK-BASED EVALUATION OF 
FLOOD WARNING AND 
PREPAREDNESS SYSTEMS 
Volume 1 - Overview 




IWR Report 95-R-6 
December 1995 


20100622238 










RISK-BASED EVALUATION OF 
FLOOD WARNING AND 
PREPAREDNESS SYSTEMS 
Volume 1 - Overview 


by 

Yacov Y. Haimes, Ph.D., P.E., Project Director 
Duan Li, Ph.D. 

Vijay Tulsiani, M.S. 

James H. Lambert, Ph.D. 

Roman Krzysztofowicz, Ph.D., Consultant 



DECEMBER 1995 


IWR REPORT 95-R-6 




Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Preface 

This report is a product of the U.S. Army Corps of Engineers’ Risk Analysis for Water Resources 
Investments Research Program. The program is managed by the Institute for Water Resources, which is 
a unit of the Water Resources Support Center. The report was prepared to fulfill part of several work units 
in the research program. These work units focused on developing and applying the concepts of risk 
preference and risk communication to water resources issues. The report conforms to the basic planning 
model and to the risk and uncertainty analysis recommendations presented in Economic and Environmental 
Principles and Guidelines for Water Related Land Resources Implementation Studies (P&G). 

The risk analysis framework encompasses the four basic steps in dealing with any risk: 
characterization, qualification, evaluation, and management. The purpose of conducting these analyses is 
to provide additional information to both Federal and non-Federal partners on the engineering and economic 
performance of alternative investments that address water resources problems. The goal is to produce better 
informed decisions and to foster the development of the idea of rational joint consent by all parties to an 
investment decision. 

This report, entitled Risk-Based Evaluation of Flood Warning and Preparedness Systems, represents 
a synthesis and elaboration of three earlier technical reports 1 to the Institute for Water Resources prepared 
by Environmental Systems Modeling, Inc. The results presented here have as a unifying theme that design 
and evaluation of structural and nonstructural measures for flood mitigation, including flood warning and 
preparedness systems, is an integrative, holistic process that requires an understanding of the contribution 
each type of measure makes to the performance of the overall system. The models rely on concepts of 
multiobjective decisionmaking, tradeoff analysis, and the risk of extreme events. This report is divided into 
OVERVIEW and TECHNICAL sections. Each of the four OVERVIEW sections summarizes in a 
nontechnical style a methodology developed for the integration of flood warning and preparedness systems 
into the design and evaluation process. The four methodologies are (1) integration of structural measures 
and flood waming/preparedness systems, (2) multiobjective decision-tree analysis, (3) performance 
characteristics of a flood warning system, and (4) selection of optimal flood warning threshold. Each 
OVERVIEW section describes the main features of the model, case study, or example. The four 
TECHNICAL sections correspond to the sections of the OVERVIEW and contain the mathematical details 
that would be needed in an application of the methodologies. In addition to being a consultant for this 
report, Prof. Roman Krzysztofowicz is the sole author of the TECHNICAL section of Part 3-Performance 
Characteristics of a Flood Warning System; the OVERVIEW section of Part 3 is excerpted and edited from 
the same TECHNICAL section. The contribution and description of case-study data in Section 4—Selection 
of Optimal Flood Warning Threshold—also is adopted from work of Krzysztofowicz. 


‘Performance Characteristics of a Flood Warning System and Selection of Optimal Warning 
Threshold (September 1990); Case Studies in Selecting Optimal Flood Warning Threshold (March 
1992); and Integration of Structural Measures and Flood Warning Systems for Flood Damage 
Reduction (March 1992) 


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Flood Warning and Preparedness Systems 


Mr. Stuart A. Davis, of the Institute for Water Resources’ Technical Analysis and research 
Division, provided technical review for this report. Dr. Eugene Z. Stakhiv, Chief of the Policy and Special 
Studies Division, reviewed the earlier technical reports on which this report is based. Dr. David A. Moser, 
of the Technical Analysis and Research Division, is the principal investigator for the Risk Analysis 
Research Program. The Chief of the Technical Analysis and Research Division is Mr. Michael R. Krouse, 
and the Director of the Institute for Water Resources is Mr. Kyle E. Schilling. Mr. Robert M. Daniel, Chief 
of the Economics and Social Analysis Branch, Planning Division; Mr. Earl E. Eiker, Chief of the 
Hydrology and Hydraulics Branch, the Operations, Construction, and Readiness Division at the 
Headquarters of the U.S. Army Corps of Engineers serve as technical monitors for the research program. 
Numerous field reviewers provided valuable insights and suggestions to improve early drafts. 


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Table of Contents 


Preface.iii 

Introduction. 1 

Multiple Objectives. 1 

Impact Analysis . 3 

The Risk of Extreme and Catastrophic Events. 3 

Flood Forecasting and Waming/Preparedness Systems.4 

Toward Implementation of the Methodologies.4 

Organization of the Report .5 

Part 1 - Integration of Structural Measures and Flood Warning Systems .7 

Introduction.7 

Features of the Model .8 

Case Study-Moorefield, WV. 10 

Local Flood Protection at South Fork and South Branch Potomac Rivers at 

Moorefield, West Virginia. 10 

Tradeoff Analysis. 11 

Part 2 - Multiobjective Decision-Tree Analysis . 15 

Introduction. 15 

Overview . 15 

Multiple Objectives. 15 

Impact Analysis. 16 

The Risk of Extreme and Catastrophic Events. 16 

Methodological Approach. 17 

Extension of the Decision Tree to Multiple Objectives . 17 

Impact of Experimentation. 19 

Extension of the Decision Tree to Multiple Risk Measures . 19 

Conclusions. 21 

Example Problem for the Discrete Case. 22 

Problem Definition. 22 

Summary of the Results. 24 

Example Problem for the Continuous Case. 26 

Problem Definition. 26 

Summary of the Results. 26 

Part 3 - Performance Characteristics of a Flood Warning System. 37 

Introduction. 37 

Features of the Model . 37 

System Model . 37 


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Risk-Based Evaluation of 

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Performance Measures. 39 

Closure . 42 

Case Study-Milton, Pennsylvania. 42 

General Description . 42 

Input Models and Parameters . 42 

Properties of the ROC and the PTC . 46 

Performance Differences Between Zones. 46 

Operating Points. 48 

Performance Tradeoffs . 50 

Part 4 - Selection of Optimal Flood Warning Threshold. 57 

Introduction. 57 

Features of the Model . 57 

Case Studies. 61 

Application to Milton, Pennsylvania. 61 

Application to Eldred, Pennsylvania. 67 

Application to Connellsville, Pennsylvania. 71 

GLOSSARY OF SYMBOLS. 77 

REFERENCES. 85 

List of Figures 

Figure 1-1. Curves for Relating Flood Frequency, Discharge, Elevation, and Damage .9 

Figure 1-2. Cost vs. Damage Tradeoff for the Expected Value (f 5 ) . 12 

Figure 1-3. Cost vs. Conditional Expected Damage (f 4 ) Tradeoff for a = 0.9 . 13 

Figure 1-4. Cost vs. Conditional Expected Damage (f 4 ) Tradeoff for a = 0.99 . 13 

Figure 1-5. Tradeoffs among Cost, Expected Damage (f 5 ), and Conditional Expected Damage (f 4 ) 

for the Four Alternatives. 14 

Figure 2-1. Structure of Multiobjective Decision Trees. 18 

Figure 2-2. Re-shape of the Feasible Region by an Experimentation . 20 

Figure 2-3. Variance and Region of Extreme Events. 21 

Figure 2-4. Decision Tree for the Discrete Case . 23 

Figure 2-5. Decision Tree for the First Stage (Discrete Case). 28 

Figure 2-6. Pareto Optimal Frontier (Discrete Case). 29 

Figure 2-7. Decision Tree for the Continuous Case . 30 

Figure 2-8. Decision Tree for the Second Stage Using f 5 (Continuous Case) . 35 

Figure 2-9. Pareto Optimal Frontier Using f 5 (Continuous Case). 36 

Figure 3-1. Functional Structure of a Flood Warning System. 38 

Figure 3-2. Tree of Events Leading to One of the Four Performance States of a Flood Warning 

System . 41 


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Figure 3-3. Expected Lead Time of a Flood Warning Versus the Elevation of the Floodplain 

for Three Warning Systems in Milton, Pennsylvania. 44 

Figure 3-4. Relative Operating Characteristics (ROC) of Warning System SI for Four Zone 

Elevations, y, in Milton, Pennsylvania. 47 

Figure 3-5. Performance Tradeoff Characteristics (PTC) of Warning System SI for 

Four Zone Elevations, y, in Milton, Pennsylvania. 49 

Figure 3-6. Relative Operating Characteristics (ROC) of Three Warning Systems, SI, S2, 

and S3 for Zone Elevation y = 22 ft in Milton, Pennsylvania. 51 

Figure 3-7. Performance Tradeoff Characteristics (PTC) of Three Warning Systems, 

SI, S2, and S3, for Zone Elevation y = 22 ft in Milton, Pennsylvania. 52 

Figure 3-8. Relative Operating Characteristics (ROC) of Three Warning Systems, SI, S2, 

and S3, for Zone Elevation y = 28 ft in Milton, Pennsylvania. 53 

Figure 3-9. Performance Tradeoff Characteristics (PTC) of Three Warning Systems, SI, S2, 

and S3, for Zone Elevation y = 22 ft in Milton, Pennsylvania. 54 

Figure 4-1. Multilevel Structure of Flood Warning Systems. 58 

Figure 4-2. Interaction Between Forecast and Response Subsystems. 60 

Figure 4-3. Trade-off Between Type I and Type II Errors. 63 

Figure 4-4. Relationship Between the Expected Flood-loss Reduction and the Warning 

Threshold for Various Response Fraction and Elevation Levels. 65 

Figure 4-5. Relationship Between the Optimal Warning Threshold and the Weighting 

Coefficient for a = 0.55 at Stage 4. 68 

Figure 4-6. Relationship Between the Optimal Warning Threshold and the Stages for a = 0.70 

and 0 = 0.1. 69 

Figure 4-7. Relationship Between the Optimal Warning Threshold and the Response Fraction 

for 0 = 0.02 at Stage 3 . 70 

List of Tables 

Table 1. Assumptions, Main Functions, and Limitations of the Four Methodologies.2 

Table 1-1. Summary of Results: Tradeoffs Among Cost, Expected Damage (f 5 ), and Risk 

of Extreme Events (f 4 ) for the Four Alternatives . 11 

Table 2-1. Expected Value of Loss Vectors for the Second-period Decision Arcs 

(Discrete Case). 25 

Table 2-2. Noninferior Decisions for the Second-period Decision Nodes (Discrete Case) . 25 

Table 2-3. Decisions for the First-Period Decision Node (Discrete Case). 27 

Table 2-4. Loss Vectors for the Second-period Decision Arcs (Continuous Case) . 31 

Table 2-5. Noninferior Decisions for the Second-period Decision Nodes (Continuous Case) ... 31 

Table 2-6. Decisions for the First-period Decision Node Using f 5 (Continuous Case). 33 

Table 2-7. Decisions for the First-period Decision Node Using f 4 (Continuous Case). 34 

Table 3-1. Parameters of Three Alternative Designs of a Flood Warning System 

for Milton, Pennsylvania . 45 

Table 3-2. Expected Number of Zone Floods and Expected Lead Times of Flood Warnings 

for Milton, Pennsylvania . 45 


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Table 3-3. Coordinates of Operating Points on the ROC and PTC Curves That Give 

the Same Probability of False Warning P(F) for Each Zone Elevation; System 

Design SI for Milton, Pennsylvania . 48 

Table 3-4. Coordinates of Three Alternative Points on the ROC and PTC Curves for Zone 

Elevation y = 22 ft. 54 

Table 4-1. Probabalistic Measures of the Warning System . 62 

Table 4-2. Expected Flood-loss Reduction. 64 

Table 4-3. Probabilistic Measures of the Warning System (Connellsville, Pennsylvania). 74 


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Flood Warning and Preparedness Systems 


Introduction 



Each of the four methodologies developed in this report contributes an important dimension to 
risk-based evaluation of systems for flood damage reduction — which is incomplete without 
accounting for both structural and nonstructural measures. The unifying theme of these results is 
that the design and evaluation of structural and nonstructural measures for flood mitigation, 
including flood warning and preparedness systems, is an integrative, holistic process that 
eventually must build on an understanding of the contribution of each type of measure to the 
performance of the overall system. Furthermore, the design of flood mitigation is tied to multiple 
objectives of minimizing cost and risk and maximizing performance. Consideration of the risk of 
extreme events is an essential element in the evaluation of design tradeoffs. 

The four methodologies developed here for the modeling and evaluation of flood warning and 
preparedness systems are: 

(1) Integration of structural measures and flood waming/preparedness systems, 

(2) Multiobjective decision-tree analysis, 

(3) Performance characteristics of a flood warning system, and 

(4) Selection of optimal flood warning threshold. 

The assumptions, main functions, and limitations of the four methodologies are summarized in 
Table 1. 

Multiple Objectives 

The single-objective models that had been advanced in the fifties, sixties, and seventies are today 
considered by many to be unrealistic, too restrictive, and often inadequate for most real-world 
complex problems. The proliferation of books, articles, and conferences and courses during the 
last decade or two on what has come to be known as multiple-criteria decisionmaking (MCDM) 
is a vivid indication of this somber realization and of the maturation of the field of decisionmaking 
[see Chankong and Haimes 1983]. In particular, an optimum derived from a single-objective 
mathematical model, including that which is derived from a decision tree, often may be far from 
representing reality — thereby misleading the analyst(s) as well as decisionmaker(s). 
Fundamentally, most complex problems involve, among other things, the minimization of costs, 
the maximization of benefits (not necessarily in monetary values), and the minimization of risks 


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Risk-Based Evaluation of 

Hood Warning and Preparedness Systems 


Table 1. Assumptions, Main Functions, and Limitations of the Four Methodologies 



Assumptions 

Main Functions 

Limitations 

Integration of structural 
measures and flood 
warning/ 

preparedness systems 

Knowledge of flood 
frequency, discharge, 
stage, and damage 
relationships for various 
combinations of structural 
and flood 

waming/preparedness 

systems. 

Determine the optimal 
design options from 
among alternative 
combinations of structural 
and flood 

waming/preparedness 
measures in a 
multiobjective framework, 
including cost, the 
expected flood loss, and 
risk of extreme floods . 

No operational issues 
associated with 
waming/preparedness 
systems and structural 
measures are considered. 

Multiobjective decision- 
tree analysis 

Knowledge of the 
probabilities for the 
underlying distributions of 
water level. Knowledge of 
severity of loss with 
alternative decisions at 
various time stages. 

Determine the optimal 
sequential decisions in an 
individual flood event 
based on the observation 
of water stage. 

No flood forecast is taken 
into consideration. 

Performance 
characteristics of a flood 
warning system 

Knowledge of the joint 
probability description of 
flood forecast and actual 
flood crest. 

Provide an evaluation 
model of the performance 
of a flood forecast system. 
In particular, the ROC 
curve characterizes the 
tradeoff between the 
probabilities of detection 
and false warning. 

Interactions between 
successive flood events 
through the dynamics of 
the community response 
fraction are not taken into 
account. 

Selection of optimal flood 
warning threshold 

Knowledge of the joint 
probability description of 
flood forecast and actual 
flood crest. Knowledge of 
the loss to the community 
associated with flood 
stage. Knowledge of the 
dynamics of the 
community response 
fraction. 

Find the optimal threshold 
level at which to issue a 
flood warning in order to 
balance the desire for high 
present-flood*loss 
reduction with the 
possibility of high future 
flood loss being 
inevitable. 

The derived optimal 
threshold may not be 
stationary; i.e., the 
optimal threshold may 
vary in different flood 
events even if the 
community response 
fraction is the same. 


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Risk-Based Evaluation of 
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of various kinds. For example, decision trees, which are a powerful mechanism for the analysis 
of complex problems, can better serve both the analysts and the decisionmakers when they are 
extended to deal with the above multiple objectives. 

Impact Analysis 

On a long-term basis, managers and other decisionmakers are often rewarded not because they have made 
many optimal decisions in their tenure; rather, they are honored and thanked for avoiding adverse and 
catastrophic consequences. If one accepts this premise, then the role of impact analysis -- studying and 
investigating the consequences of present decisions on future policy options -- might be as important, if not 
actually more so, than generating an optimum for a single-objective model or identifying a Pareto-optimum 
set (a Pareto-optimum, or non-inferior, alternative cannot be improved in any one objective without seeing 
a corresponding loss with respect to one or more other objectives) for a multiobjective model. Certainly, 
when the ability to generate both is present, having an appropriate Pareto-optimum set and knowing the 
impact of each Pareto-optimum on future policy options should enhance the overall decisionmaking 
process. 

The Risk of Extreme and Catastrophic Events 

Risk, which is a measure of the probability and severity of adverse effects, has until recently been 
commonly quantified via the expected-value formula. This formula essentially precommensurates events 
of low frequency and high damage with events of high frequency and low damage. Although learned 
students of risk analysis recognize the disparity between the above fallacious representation of extreme and 
catastrophic events and the perception of these events by individuals or the public at large, many continue 
to use this approach. The trend, however, is moving toward the conditional-expected-value approach, where 
extreme and catastrophic events are partitioned, isolated, quantified in terms of conditional expectation 
(e.g., using concepts from the statistics of extremes), and then evaluated along with the common expected 
value of risk or damage [Asbeck and Haimes 1984; Haimes 1988; Karlsson and Haimes 1988]. 

The partitioned multiobjective risk method (PMRM) developed by Asbeck and Haimes [1984] separates 
extreme events from other noncatastrophic events, and thus provides the decisionmaker(s) with additional 
valuable and useful information. In addition to using the traditional expected value, the PMRM generates 
a number of conditional expected-value functions, termed here risk functions, which represent the risk, 
given that the damage falls within specific probability ranges (or damage ranges). 

Combining either a conditional expected risk function or the unconditional expected risk function with the 
cost objective function creates a set of multiobjective optimization problems in which the tradeoffs between 
cost and the risk arising from the various ranges of damage are analyzed. This formulation offers more 
information about the probabilistic behavior of the problem than the single multiobjective formulation that 
minimizes only the cost and the expected damage. The tradeoffs between the cost function and any risk 
function allow decisionmakers to consider the marginal cost of a small reduction in the risk objective, given 
a particular level of risk assurance for each of the partitioned risk regions, and given the unconditional risk 
function. 


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Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


Flood Forecasting and Warning/Preparedness Systems 

Flood control can be provided by either structural or nonstructural measures or a combination of 
both. Structural flood control measures, such as an increase in dam height, affect the flood-frequency 
relationship. Nonstructural measures, such as a flood waming/preparedness system, do not have an impact 
on the flood-frequency relationship; however, they modify the flood-damage relationship. 

The benefits of flood forecasts have been studied and systems approaches to flood forecasting have 
been pursued by many research scholars for more than twenty years [NACOA 1972; Bhavnagri and 
Bugliarello 1965; Bock and Hendrick 1966; Day and Lee 1976; Lee et al. 1975; Sniedovich et al. 1974; 
Sniedovich and Davis 1977]. Curtis and Schaake [1988] evaluated flood warning benefits both on a national 
(or regional) scale and on a specific site problem. Prediction models for loss of life from floods were 
studied by Lee et al. [1986] and Shabraan [1987], Barrett et al. [1988] developed categories for flood 
warning systems based on types of flood forecasting systems and flood response systems. 

Predicting the future behavior of a time-dependent random variable is a major research task in the 
theory and applications of stochastic processes. Criucal events occur when the level of the random variable 
crosses a given high level (e.g., flooding level). An alarm is set off when the random variable exceeds a 
specified threshold level. An alarm system is considered optimal if it detects catastrophes with an acceptable 
level of probability and at the same time yields a minimum expected number of false alarms [Lindgren 
1979, 1980, 1985; de Mar6 1980]. The paper by de Mar6 [1980] indicates that when judging the 
performance of an alarm system, it is not very interesting to know, in the mean, how close the prediction 
is to the actual process; however, it is important for a system to be able to detect catastrophes without 
causing too many false alarms. 

In a series of papers, Krzysztofowicz and his colleagues [Alexandridis and Krzysztofowicz 1985; 
Ferrell and Krzysztofowicz 1983; Krzysztofowicz 1983a, b; 1985; Krzysztofowicz and Davis 1983a, b, 
c, d; 1984] conceptualized the flood forecast-response process in the form of a total system. This system 
is defined as a cascade coupling of two components: (1) the forecasting system, which includes data 
collection, flood forecasting, and forecast dissemination; and (2) the response system, which encompasses 
decisionmaking and action implementation. Based on the above mathematical description of the physical 
flood forecast-response process, Krzysztofowicz and his colleagues establish performance measures of flood 
warning systems. 

Pat6-Comell [1986] presents a method for assessing the performance of the forecasting system and 
human response, given the memory that people have kept on the quality of previous alerts. The tradeoff 
between the rate of false alerts and the length of the lead time is studied to account for the long-term effects 
of "crying wolf." An explicit formulation of benefits from warning systems is derived under the above 
considerations. 

Toward Implementation of the Methodologies 

An immediate and most worthwhile challenge is the refinement of the four methodologies of this 
report for the operational setting. For instance, a decision support system for the risk-based evaluation of 


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Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


flood warning systems might be developed to integrate these methodologies in a framework consistent with 
Corps of Engineers planning procedures [HEC 1988]. 

Organization of the Report 

The body of this report has two volumes: OVERVIEW (Volume 1) and TECHNICAL (Volume 
2). Each of the four sections in the OVERVIEW Volume summarizes in a nontechnical style a 
methodology developed for the integration of flood warning systems into the design and evaluation process. 
The OVERVIEW Volume describes the main features of the model, case study, or example. The four 
TECHNICAL Volume sections correspond to the sections of the OVERVIEW and contain the mathematical 
details that would be needed in an application of the methodologies. The OVERVIEW’S present an 
excerpted group of the figures and tables used in the TECHNICAL sections. 


5 




Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Part 1 

Integration of Structural Measures 
And Flood Warning Systems: Overview 

Introduction 

In most cases, the maximum flood loss reduction can be only achieved through an optimal 
combination of both structural and nonstructural flood control measures, since the adoption of integrated 
measures will certainly enlarge the feasible region of flood control measures in comparison with situations 
where only structural or nonstructural measures alone are considered. Structural measures include the 
construction of reservoirs, levees, and flood walls. Nonstructural measures include floodplain land use 
planning, flood insurance, flood warning systems, floodproofing, and permanent relocation. Various flood 
control measures prevent inundation of the floodplain in different ways and have different impacts on the 
flood damage-frequency relationship. A structural measure, such as an increase in reservoir height, affects 
the frequency-discharge relationship; levees and flood walls confine the discharge within certain channels, 
thus changing the relationship between discharge and elevation; most nonstructural measures, such as a 
flood warning system, modify the stage-damage relationship. 

The idea of combining both structural and nonstructural measures in flood control is not new. 
Various research results have been reported in which structural measures are combined with nonstructural 
measures, such as zoning, floodproofing, and flood insurance. Readers can refer to Thampapillai and 
Musgrave [1985], which provides a comprehensive survey in reviewing integrated structural and 
nonstructural measures in flood damage mitigation. To date, however, no other research work on 
combining structural measures with flood warning systems has appeared in the literature. 

Issues of both design and operation are involved in structural measures as well as nonstructural 
ones. Building a reservoir is a structural measure in flood control. Determination of the height of the 
reservoir is a design issue, while determination of the amount of the release on a monthly or daily basis is 
an operational issue. Installing a flood warning system is a nonstructural measure in flood control. 
Determination of an acceptable reliability of a warning system is a design issue with a consideration of the 
system cost, while determination of the flood warning threshold for various flood events is an operational 
issue. It is important to note that operational issues can only be addressed in a framework of dynamic 
optimization. For example, different levels of flood warning thresholds will cause different probabilities 
of missed forecast and false alarm, thus affecting the fraction of the community's future response. In this 
part, we analyze design options for the combination of structural measures and flood warning systems, thus 
building on and extending the existing methodology of computing flood loss for a given structural measure 
developed by the Army Corps of Engineers to incorporate flood warning systems. 

For the computation of flood loss for a given flood-control structural measure, a widely-used 
procedure developed by the Army Corps of Engineers investigates the relationships between discharge vs. 
frequency, discharge vs. elevation, and damage vs. elevation, such that the damage-frequency curve can 
be generated for an average annual flood loss. An integrative approach is developed in this part to combine 
the calculation of flood loss reduction through flood warning systems with the calculation of flood loss for 
a given flood-control structure, thus facilitating the evaluation of combined structural measures and flood 
warning systems in reducing flood loss. Recognizing the inadequacy of the expected value as the sole 



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Risk-Based Evaluation of 

Flood Wanting and Preparedness Systems 


measure of risk, the partitioned multiobjective risk method (PMRM), which was employed in an earlier 
Army Corps of Engineers study on dam safety and which provides an added risk measure of extreme 
events, is adopted in this study. The conventional expected value of loss, the conditional expected value of 
extreme loss generated by the PMRM, and the cost associated with the structural measures and flood 
warning systems both separately and together are traded-off and analyzed in a multiobjective framework. 
An example problem that uses data from case studies performed for or by the U.S. Army Corps of 
Engineers demonstrates the efficacy and contribution of the developed integrated methodological 
framework. In particular, integrated structural measures for flood control and flood warning systems 
demonstrate clear and undisputed advantages over each system separately. The integrated system also 
provides a wider range of alternatives that makes flood protection more cost effective. The developed 
integrated methodological framework is simple to understand and use, since it builds on existing Army 
Corps of Engineers practices and uses accepted and already adopted procedures. The new methodology, 
however, brings an additional measure of risk to the Corps analyses -- the risk of extreme events in a 
multiobjective framework — and thus makes the entire analysis more comprehensive and meaningful for 
decisionmaking purposes. 


Features of the Model 

In the procedure for computing flood damage that has been developed by the Army Corps of 
Engineers, four functional relationships (or curves - Fig. 1-1) are needed to completely quantify each 
alternative of structural measures: 

1) curve of frequency vs. discharge, 

2) curve of elevation vs. discharge, 

3) curve of elevation vs. damage, and 

4) curve of frequency vs. damage. 

Note here that the fourth curve can be derived if the other three are known. In general, the relationships 
of frequency vs. discharge, elevation vs. discharge, and damage vs. elevation are constructed from real data 
such that the curve of frequency vs. damage can be derived in order to compare the expected flood damage 
for structural measures. The approach of discrete enumeration of all possible combinations of both 
structural and nonstructural measures is adopted in our development. 

In this report we subscribe to a premise that the introduction of a flood warning system will not 
affect the relationships between the frequency and discharge and between the elevation and discharge. It 
will, however, alter the curve of elevation vs. damage, thus changing the relationship between frequency 
and damage. 

To evaluate the flood loss reduction by installing a flood warning system, the concept of category-unit 
loss function detailed by Krzysztofowicz and Davis [1983] is adopted. 


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Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 



Figure 1-1. Curves for Relating Flood Frequency, Discharge, Elevation, and Damage 

One of the critical parameters used in this study to evaluate flood loss is the fraction of people in 
a community who respond to flood warnings. The cost function of evacuation is assumed to be a linear 
function of the response fraction. 

The flood loss function without a warning system is essentially given by the curve of elevation vs. 
damage (Figure 1-lc) for each given structural measure. Alternatively, the flood loss function without a 
warning system can be defined by reference to a unit damage function specifying the fraction of the 
maximum possible damage that occurs for a given depth of flooding. 

The flood loss with a warning system is taken to be a function of all of the parameters considered 
in the case of no warning system and, additionally, the evacuation costs of the response fraction of the 
community less the losses avoided by an evacuation. 

The value of the maximum flood loss for a community and the functional form of the unit damage 
function can be obtained from the curve of elevation vs. damage when structural measures are evaluated. 
The only additional information required to calculate the relationship of elevation vs. flood loss reduction 
through a flood warning system are the value of maximum evacuation cost, the value of response fraction 
in the community and the losses avoided by an evacuation. The resulting curve of elevation vs. flood loss 


9 




















Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


reduction can be viewed as a function parametrized by the response fraction. We should note, however, 
that the flood loss reduction is a linear function of the response fraction 0. 

Reducing the value of damage in the curve of damage vs. elevation for each structural measure by 
a function of the elevation and the response fraction yields a new relationship between elevation and damage 
when a flood warning system is introduced. Setting the response fraction equal to one yields a maximum 
achievement of flood loss reduction. Combining this new curve of elevation vs. damage with other two 
curves of frequency vs. discharge and elevation vs. discharge provides us with a new relationship between 
frequency and damage for a combined structural measure and a flood warning system. 

Although the relationship of damage vs. frequency provides the most complete evaluation for each 
flood control alternative, it is necessary to compress information to generate a risk measure when various 
flood control alternatives are compared. The most commonly used risk measure is the expected value of 
the flood loss. Although the expected-value approach indicates the central tendency of flood damage for 
each flood control alternative, it fails to separate extreme, catastrophic flood events from the rest. The 
partitioned multiobjective risk method (PMRM) [Asbeck and Haimes 1984] adopts the concept of 
conditional expectation, which enables us to isolate, quantify, and evaluate the impact of each flood control 
alternative on extreme, catastrophic flood events. 

Multiobjective analysis is performed to evaluate the various flood control alternatives. In the 
PMRM [Asbeck and Haimes 1984], for each flood control alternative we calculate three objective functions: 
the cost, the expected value of damage, and the conditional expectated value of flood damage from extreme 
floods whose return periods are greater than a preset value. Both the expected damage and the conditional 
expected damage can be derived from the curve of damage vs. frequency for each flood control alternative. 

The conditional expected damage gives information about the risk of extreme events that is 
overlooked in analyses where only the expected value of damage is considered. A flood control decision 
may have different impacts on the expected flood loss and the conditional expected flood loss from floods 
whose return period exceed a certain threshold level. The evaluation of risk in the multiobjective framework 
of the PMRM provides more decision aids to determine the optimal flood control strategy. The added 
tradeoff information between the cost and the risk of extreme flood loss addresses explicitly the public 
concern about catastrophic flood loss. The importance of including information on the risk of extreme 
events will be demonstrated through the multiobjective tradeoff analysis presented in the case study below. 


Case Study-Moorefield, WV 

This section develops an example to illustrate the integrated approach described in the previous 
section. A study undertaken at the South Fork and South Branch Potomac Rivers at Moorefield, West 
Virginia, in 1990 was selected as the basis for the development of the example. 

Local Flood Protection at South Fork and South Branch Potomac Rivers at Moorefield, West Virginia 

The documents provided for this study were a reconnaissance report dated September 1987 and an 
Integrated Feasibility Report and Environmental Impact Statement dated March 1990. The latter consists 
of a main report and 13 appendices [U.S. Army Corps of Engineers 1990]. 


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Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


The following extracts from the main report [U.S. Army Corps of Engineers 1990] provide some 
background for this example: 

The town of Moorefield in Hardy County, West Virginia, is subject to flooding from the 
South Fork and South Branch Potomac River. Serious floods have occurred in March 
1936, June 1949, and November 1985 . 

In response to the flooding problem, the Corps of Engineers and the Interstate Commission 
on the Potomac River Basin initiated a cost-shared feasibility study in February 1988 to 
identify and evaluate possible solutions. 

A range of possible structural and nonstructural measures was examined. These measures 
included levees, floodwalls, channel improvements, bridge modification, and nonstructural 
alternatives. The most effective measures were combined into plans for comparison to the 
without project condition. 

In the example developed here, two given structures of flood control will be investigated. Plan 1 is the 
zero-cost plan, that is, the without-project-condition alternative. Plan 4 is a structural plan and includes levees 
and floodwalls to protect residential areas, industrial plants, businesses, schools, and commercial areas in both 
North and South Moorefield. A detailed description of this plan is provided on page 67 of the main report 
[U.S. Army Corps of Engineers 1990]. More details about the two plans can be found in Part 1 of the technical 
volume. 

Tradeoff Analysis 

Once the conditional and unconditional expected values for the different plans are computed as 
described in the technical section, we can perform a tradeoff analysis in terms of costs and damages. The 
compiled results are shown in Table 1-1. 


Table 1-1. Summary of Results: Tradeoffs Among Cost, Expected Damage (f 5 ), and Risk of Extreme 

Events (f 4 ) for the Four Alternatives 



Average 
Annual Cost 
($ million) 

f,(L) 

f„(L|a=0.9) 

f 4 (L| 0=0.99) 

Plan 1 

0.000 

0.377 

3.428 

6.657 

Plan 1+W 

0.050 

0.159 

1.586 

4.086 

Plan 4 

0.865 

0.204 

2.515 

5.705 

Plan 4 + W 

0.915 

0.084 

1.412 

3.348 


11 




















Risk-Based Evaluation of 

Flood Warning and Pr iredrtess Systems 


. ce plan 1 is the option of doing nothing, it does not have an associated cost. The average annual cost 
for plan 4 is given as $0,865 million for a 50-year level of protection [Table 1-18, page 1-29, U.S. Army 
Corps of Engineers 1990]. The average annual cost of the flood warning system is assumed to be $50,000. 
Figure 1-2 shows the resulting tradeoffs when using the expected value alone. The solid line shows the 
Pareto optimal frontier. (An option is Pareto optimal, or noninferior, if there is no other option that 
improves on any particular objective function without loss of ground in the other objective functions.) 
Figures 1-3 and 1-4 show the tradeoffs between annual cost and flood losses for a = 0.9 and 0.99 
respectively, the a’s corresponding to two levels of the risk of extreme events. Figure 1-5 shows these same 
tradeoffs together. In these figures, f 5 denotes the expected value of loss and f 5 denotes the conditional 
expected loss, a measure of the risk of extreme events. As explained in the technical volume, the conditional 
expected loss f 4 with a = 0.9 is the average loss over the worst ten percent of floods; similarly, f 4 with a 
= 0.99 is the average loss over the worst one percent of floods. Note from Figure 1-5 that the option of 
plan 4 without the warning system is noninferior, or a viable option, in considering only structural 
measures; the same option becomes inferior when considering the warning system options. Plan 1, plan 
1+W, and plan 4+W constitute the noninferior set of options in the combined structural/nonstructural 
analysis. The combined analysis of structural and nonstructural measures, incorporating the risk of extreme 
events, clearly demonstrates the relative inefficiency of plan 4 without the warning system, an important 
result that would not have come from a traditional approach. 



Figure 1-2. Cost vs. Damage Tradeoff for the Expected Value (f 3 ) 
(Note that there are three Pareto optimal (efficient) alternatives) 


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Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 



Figure 1-3. Cost vs. Conditional Expected Damage (f 4 ) Tradeoff for a = 0.9 
(Note the three pareto optimal (efficient) alternatives) 



Figure 1-4. Cost vs. Conditional Expected Damage (f 4 ) Tradeoff for a = 0.99 
(Note the three Pareto optimal (efficient) alternatives) 


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Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


• /5(L) 

o /4(U o=0.9) 



Damages ($ million) 


Figure 1-5. Tradeoffs among Cost, Expected Damage (fj), and Conditional Expected Damage (f„) for 
the Four Alternatives: Risk of Extreme Events (f 4 ) Evaluated at Two Partitioning Levels (a = 

0.9, a = 0.99) 


14 















Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Part 2 

Multiobjective Decision-tree 
Analysis: Overview 

Introduction 

Overview 

Decision-tree analysis, which is especially useful when related actions are taken in a sequence, has 
emerged over the years as an effective and useful tool in decisionmaking. More than two decades ago, 
Howard Raiffa [1968] published the first comprehensive and authoritative book on decision-tree analysis. 
Ever since, its applications to a variety of problems from numerous disciplines have grown by leaps and 
bounds [see Sage 1977 and Hamburg 1970]. Advances in science and in scientific approaches to problem 
solving are often made on the basis of earlier works of others. In this case, the foundation for Raiffa's 
contributions to decision tree analysis can be traced to the works of Bernoulli on utility theory [see von 
Neumann and Morgenstem 1944; Edwards 1967; Savage 1954; Adams 1960; Arrow 1963; Shubik 1964; 
Luce and Suppes 1965; and others]. This chapter, in an attempt to build on the above seminal works, 
extends and broadens the concept of decision-tree analysis to incorporate: (a) multiple, noncoraraensurate 
and conflicting objectives, (b) impact analysis, and (c) the risk of extreme and catastrophic events [Haimes 
et al. 1990], In contrast, the current practice often involves one-sided use of decision trees -- optimizing 
a single objective function and commensurating infrequent catastrophic events with more frequent 
noncatastrophic events using the common unconditional mathematical expectation. 

We consider below (see Section 2.3) the application of multiobjective decision trees to a flood- 
warning and evacuation problem. Three possible actions, (a) evacuation, (b) issuing a flood watch, and (c) 
doing nothing, are under consideration at each of two decision periods. There are costs associated with the 
first two options. Furthermore, the cost associated with each option is a function of the period in which the 
action is taken. The evaluative performance measures used include the expected and conditional expected 
costs (losses) and the expected and conditional expected loss of lives resulting from a series of flood 
watch/evacuation decisions. The conditional expected value (of cost and of lives lost) is a measure of the 
risk of extreme events that is generated in the partitioned multiobjective risk method (PMRM) described 
below. 



Multiple Objectives 

The single-objective models that had been advanced in the fifties, sixties, and seventies are today 
considered by many to be unrealistic, too restrictive, and often inadequate for most real-world complex 
problems. The proliferation of books, articles, and conferences and courses during the last decade or two 
on what has come to be known as multiple-criteria decisionmaking (MCDM) is a vivid indication of this 
somber realization and of the maturation of the field of decisionmaking [see Chankong and Haimes 1983]. 
In particular, an optimum derived from a single-objective mathematical model, including that which is 
derived from a decision tree, often may be far from representing reality -- thereby misleading the analyst(s) 
as well as the decisionmaker(s). Fundamentally, most complex problems involve, among other things, the 
minimization of costs, the maximization of benefits (not necessarily in monetary values), and the 


IS 





Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


minimization of risks of various kinds. Decision trees, which are a powerful mechanism for the analysis 
of complex problems, can better serve both the analysts and the decisionmakers when they are extended 
to deal with the above multiple objectives. 


Impact Analysis 

On a long-term basis, managers and other decisionmakers are often rewarded not because they have 
made many optimal decisions in their tenure; rather, they are honored and thanked for avoiding adverse 
and catastrophic consequences. If one accepts this premise, then the role of impact analysis — studying and 
investigating the consequences of present decisions on future policy options - might be as important, if not 
actually more so, than generating an optimum for a single-objective model or identifying a Pareto optimum 
set for a multiobjective model. Certainly, when the ability to generate both is present, having an appropriate 
Pareto optimum set and knowing the impact of each Pareto optimum on future policy options should 
enhance the overall decisionmaking process within the decision-tree framework. 


The Risk of Extreme and Catastrophic Events 

Risk, which is a measure of the probability and severity of adverse effects, has until recently been 
commonly quantified via the expected-value formula. This formula essentially precommensurates events 
of low frequency and high damage with events of high frequency and low damage. Although learned 
students of risk analysis recognize the disparity between the above fallacious representation of extreme and 
catastrophic events and the perception of these events by individuals or the public at large, many continue 
to use this approach. The trend, however, is moving toward the conditional-expected-value approach, where 
extreme and catastrophic events are partitioned, isolated, quantified in terms of the conditional expectation 
(e.g., using concepts from the statistics of extremes), and then evaluated along with the common expected 
value of risk or damage [Asbeck and Haimes 1984; Haimes 1985; Karlsson and Haimes 1988a, 1988b; 
Haimes et al. 1988]. 

The partitioned multiobjective risk method (PMRM) developed by Asbeck and Haimes [1984] 
separates extreme events from other noncatastrophic events, and thus provides the decisionmaker(s) with 
additional valuable and useful information. In addition to using the traditional expected value, the PMRM 
generates the conditional expected-value, which represents the risk, given that the damage falls within an 
extreme range of exceedance probability (or range of damage). The conditional expected value represents 
the risk with low probability of exceedance and high damage. Combining each of the conditional expected 
value and the unconditional expected value with the competing objective of cost results in a set of 
multiobjective optimization problems from which an optimal balance of cost and risk can be found. The 
tradeoffs between the cost objective and either risk objective (conditional and unconditional expected value) 
enable decisionmakers to consider the marginal cost of a small reduction in a risk objective, given a 
particular level of risk assurance for extreme events and given the unconditional risk function. The PMRM 
is integrated with the multiobjective decision tree in the continuous-case example of Section 2.3 below. 


16 





Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Methodological Approach 

Extension of the Decision Tree to Multiple Objectives 

Similar to the decision-tree in conventional single-objective analysis, a multiobjective decision tree 
(Fig. 2-1) is composed of decision nodes, chance nodes and a time sequence of related actions and 
consequences [Haimes et al. 1990]. Following Raiffa [1968], a decision node is designated by a small 
square and a chance node by a small circle. But now each path through the tree, instead of having a single 
measure of evaluation as has been usual, is characterized by a vector-valued (multiobjective) performance 
measure. 

At a decision node, the decisionmaker selects one course of action from the feasible set of 
alternatives. We assume that there are only a finite number of alternatives at each decision node. These 
alternatives are shown as branches emerging to the right side of the decision node. The performance vector 
associated with each alternative is written along the corresponding branch. Each alternative branch may lead 
to another decision node, a chance node, or a terminal point. 

A chance node, indicates that a chance event is expected at this point; that is, one of the states of 
nature may occur. We consider two cases in this study: a) a discrete case, where the number of states of 
nature is assumed finite; and b) a continuous case, where the possible states of nature are assumed 
continuous. The states of nature are shown on the tree as branches to the right of the chance nodes, and 
their known probabilities are written above the branches. The states of nature may be followed by another 
chance node, a decision node, or a terminal point. 

Allowing for the evaluation of the multiple objectives at each decision node constitutes a significant 
extension of the average-out-and-folding-back strategy used in conventional single-objective decision-tree 
methods. The procedure for multiobjective decision trees is similar to that of a single-objective tree. At each 
decision node and at each branch emerging to the right side of the decision node, we find the corresponding 
set of vector-valued performance measures (multiple objectives) for each alternative and identify the set of 
noninferior solutions. At a noninferior solution (or Pareto optimum solution), one cannot achieve a decrease 
in any single objective without observing an increase in at least one other objective. 

In multiobjective decision-tree analysis, instead of having a single optimal value associated with a 
single-objective decision tree, we have a set of vector-valued objective values of noninferior decision 
alternatives at each decision node. In single-objective decision-tree analysis, there is no choice process at 
the chance nodes, since only an averaging-out process takes place there. In multiobjective decision-tree 
analysis, a set of Pareto optimum (noninferior) alternatives is associated with each branch emerging from 
a chance node. A vector minimization is performed to discard from further consideration the resulting 
inferior combinations. Finally, a straightforward solution technique is applied repeatedly until the set of 
noninferior solutions at the starting point of the tree is obtained. 

Together with the following comments {Impact of Experimentation and Extension to Multiple Risk 
Measures), the examples on flood watch and evacuation decisions described below provide an overview of 
the new methodology that incorporates multiple objectives and the PMRM into decision trees. 


17 





Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 



18 





















Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Impact of Experimentation 

The impact of an added piece of information (obtained, e.g., through experimentation) on different 
objectives is now addressed, and the value of the information is quantified by a vector-valued measure. In 
conventional decision-tree analysis, whether or not an experiment should be performed depends on an 
assessment of the expected value of experimentation, which is the difference between the expected loss 
without experimentation and the expected loss with experimentation. If the expected value of 
experimentation is negative, experimentation is deemed unwarranted; otherwise, the experiment that yields 
the lowest loss is selected. In multiobjective decision-tree analysis, the monetary index does not constitute 
the sole consideration; rather, the value of experimentation is judged in a multiobjective way where, in 
many cases, the noninferior frontiers generated with and without experimentation do not dominate each 
other. The added experimentation in these cases reshapes the feasible region (and thus the noninferior 
frontier) and generates new and better options for the decisionmakers (Fig. 2-2). 


Extension of the Decision Tree to Multiple Risk Measures 

Multiobjective decision-tree analysis calls for the adoption of multiple-risk measures. Often, the 
expected value, by itself, provides insufficient information for risk management. The expected value of 
adverse effects, which has been most commonly used in conventional decision-tree analysis, is in many 
cases inadequate, since this scalar representation of risk commensurates events that correspond to all levels 
of losses and to their associated probabilities. The common expected-value approach is particularly 
deficient for addressing extreme events, since these events are concealed during the amalgamation of events 
of low probability and high consequence with events of high probability and low consequence. The 
synthesis of several approaches — single-objective decision-tree analysis, multiobjective optimization, the 
partitioned multiobjective risk method (PMRM), and the statistics of extremes - has led to the development 
of the multiobjective decision-tree method. This new form of decision-tree analysis can handle different risk 
functions including the common expected value, the conditional expected value for extreme events, and the 
event with maximum probability, thus providing the decisionmaker(s) with more comprehensive knowledge 
and a robust decision policy. 

Determining the folding-back strategy associated with conditional expected values is substantially 
different from such an operadon using the conventional expected value. Unlike the latter, which is a linear 
operator, the conditional expected-value operator is nonlinear. This nonlinearity represents an obstacle in 
decomposing the overall value of the conditional expected value and in calculating it at different decision 
nodes. Thus, in calculating conditional risk functions, all performance measures at the different branches 
are mapped to the terminal points where the partitioning is performed. 

In summary, we adhere to the following rules when calculating the conditional expected value in 
the folding-back procedure of decision trees: 

1) Partition and calculate the conditional expected value of extreme damage at terminal points 
according to the conditional probability density function. 

2) Fold back and perform at each chance node the operation of the expected value. 

The justification and elaboration of these steps is given in the technical part. 


19 






Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 



Note that although reducing the variance (the uncertainty) of the risk may not contribute much to 
reducing the expected value, it often markedly reduces the conditional expected value associated with 
extreme events (see Fig. 2-3). Two benefits that result from additional experimentation include reducing 
the expected loss and reducing the uncertainty associated with decisionmaking under risk. However, in most 
cases, these two dual aspects of experimentation conflict with each other. The general framework of 
multiobjective decision-tree analysis proposed here provides a medium with which these dual aspects can 
be captured by investigating the multiple impacts of experimentation. 


20 
























































Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 



Figure 2-3. Variance and Region of Extreme Events 


Conclusions 

Multiobjective decision-tree analysis is an extension of the single-objective-based decision-tree 
analysis formally introduced more than two decades ago by Howard Raiffa [1968]. This extension is made 
possible by making a synthesis of the traditional method and more recently developed approaches used for 
multiobjective analysis and for the risk of extreme and catastrophic events. Successful applications of single¬ 
objective decision-tree analysis to numerous business, engineering, and governmental decisionmaking 
problems over the years have made the methodology into an important and valuable tool in systems 
analysis. Its extension - incorporating multiple noncommensurate objectives, impact analysis, and the 
conditional expected value for extreme and catastrophic events - might be viewed as an indicator of growth 
in the broader field of systems analysis and in decisionmaking under risk and uncertainty. Undoubtedly, 
there remain several theoretical challenges that must be addressed to fully realize the strengths and 
usefulness of the extended methodology. In this sense, the multiobjective decision-tree analysis proposed 
here constitutes the first, albeit important, step in the direction of developing improved and more 
representative models and decisionmaking tools. 


21 









Risk-Based Evaluation of 

Example Problem for the Discrete Case 

Problem Definition 

The example problem discussed here concerns a simplified flood warning and evacuation system. 
Three possible actions, (a) evacuation, (b) issuing a flood watch, and (c) doing nothing, are under 
consideration. There are cost factors associated with the first two options. The decision tree covers two time 
periods, and the cost associated with each option is a function of the period in which the action is taken. 
The complete decision tree for the problem is shown in Fig. 2-4. The following assumptions are made: 

a) There are three possible actions with associated costs for the first period: 

1) issuing an evacuation order at a cost of $5 million [EV1], 

2) issuing a flood watch at a cost of $1 million [WA1], and 

3) doing nothing at no cost [DN1]. 

b) For the second period the actions and the corresponding costs are: 

1) issuing an evacuation order at a cost of $3 million [EV2], 

2) issuing a flood watch at a cost of $0.5 million [WA2], and 

3) doing nothing at no cost [DN2]. 

c) The flood stage is reached at water flow (W) = 50,000 cfs. 

d) There are three possible underlying probability density functions (pdfs) for the water flow: 

1) W ~ lognormal (10.4,1) represented as LN,; where the parentheses designate statistical 
parameters of the lognormal distribution of flow as ( mean flow x 10 s cfs, standard 
deviation x Iff 3 cfs)', 

2) W ~ lognormal (9.1,1), represented as LN 2 , and 

3) W ~ lognormal (7.8,1), represented as LN 3 . 

The prior probabilities that any of these pdfs is the actual pdf are equal. 

e) There are four possible events at the end of the first period: 

1) A flood (W > 50,000 cfs) occurs. 


22 




Risk-Based Evaluation of 
Flood Wanting and Preparedness Systems 



10; 1400.0001 
( 0:01 

s.eoo.ooot 

10 7; 2 600.000) 
10 : 0 ) 

i (H; 3,3 00,0001 

i 10,01 


(11:1.3 

I«; 0 I 


i 10; 07 

* l U; 1.100.0 001 

* 10:07 

t <3 1:1.300 0003 

I 10:0) 

UTiMWiMM 

10:01 

i tt 4; 3.300.000) 
i 10.03 

> El i; < 300.0001 

* 10.07 

IT; T.00Q.000I 

114; 4.3 Ofl.OOOi 
1001 

, (1.1; 4.IOO.0Q01 
1 10:0) 

i It; T.OOO.OOO) 

■ 10:03 

iiH: 4 . 100.0001 
i 10.01 

i il.i: 4,ioa.ee oi 

i l Or o t 

I It; 7.O0D.000I 

i 10:01 

> 114; 4 JOO.OQOI 
i lO ;03 

■ ii.t; 4 .*eo.eoo) 
' lo.si 

it, r.ooo.oooi 
10:01 


Figure 2-4. Decision Tree for the Discrete Case 


23 


































Risk-Based Evaluation of 

Hood Warning and Preparedness Systems 


2) The water flow is greater than that of the previous period (15,000 cfs s W s 50,000 
cfs), represented as W1. 

3) The water flow is in the same range as that of the previous period (5000 cfs ^ W ^ 
15,000 cfs), represented as W2. 

4) The water flow is lower than that of the previous period (W s 5000 cfs), represented as 
W3. 

f) L = 7 and C = $7,000,000 are respectively the maximum possible loss of lives and properties, 
given no flood warning. All other costs are shown in Fig. 2-4. 


Summary of the Results 

The decision tree is simplified and solved for this example as described in the technical section. Table 
2-1 presents the values of the loss vectors for the second-period decision arcs. In folding back at each 
decision node, the vector-valued performance measures are compared to eliminate all dominated (inferior) 
policies. Consider, for example, decision node D2. The vector corresponding to the decision DN2 is 
inferior to the vector corresponding to the decision WA2 


0.3058 

1,264,579 


0.4588 

1,376,241 


where 0.3058 is the expected loss of life and $1,264,579 is the expected flood loss for the decision to issue 
a warning WA2. Note that the two evaluative measures for WA2 are each of lesser value than the 
corresponding measures for the do nothing option DN2, which is the condition for identifying an inferior 
policy. 

Table 2-2 presents the noninferior decisions for the second-period decision arcs. Averaging-out at the 
chance nodes for the first period, each noninferior decision corresponding to each arc is multiplied by the 
probability for that arc, yielding a single decision rule for the first-period decision node. For example, we 
have 18 different combinations at WA1, one of which is (EV2 | higher, EV2 | same, EV2 | lower). The 
value of the loss vector for this combination is: 


0.1779 + 0.1529 - 0.2466 ♦ 0.0704 ♦ 0.2686 + 0.0150 » 0.3577 
(1,000 + 711.76 ♦ 3,611.663 * 0.2466 ♦ 3,281.526 * 0.2686 - 3,060.043 * 0.3577)1,000 


0.2399 

4,578,391 


24 













Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Table 2-1. Expected Value of Loss Vectors for the Second-period Decision Arcs (Discrete Case) 


Node 

Arc 

L 

C 

D2 

EV2 

0.1529 

3,611,663 


WA2 

0.3058 

1,264,579 


DN2 

0.4588 

1,376,241 

D3 

EV2 

0.0704 

3,281,526 


WA2 

0.1408 

851,908 


DN2 

0.2112 

633,434 

D4 

EV2 

0.0150 

3,060,043 


WA2 

0.0300 

575,054 


DN2 

0.0450 

135,097 

D5 

EV2 

0.3058 

3,917,494 


WA2 

0.4588 

1,570,410 


DN2 

1.5292 

1,529,157 

D6 

EV2 

0.1408 

3,422,289 


WA2 

0.2112 

992,671 


DN2 

0.7038 

703,815 

D7 

EV2 

0.0300 

3,090,065 


UA2 

0.0450 

605,076 


DN2 

0.1501 

150,108 

C2 

F 

0.1779 

711,760 

C3 

F. 

0.8897 

889,700 


L — loss of lives 
C -- cost ($) 


Table 2-2. Noninferior Decisions for the Second-period Decision Nodes (Discrete Case) 


Node 

Noninferior decisions 

D2 

EV2, UA2 

D3 

EV2, WA2, DN2 

D4 

EV2, UA2, DN2 

D5 

EV2, WA2, DN2 

D6 

EV2, Wa2 , DN2 

D7 

EV2, Wa2 , DN2 


25 
















Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


where the first and second elements represent an expected loss of lives of 0.2399 and expected cost of 
$4,578,391, respectively. Table 2-3 presents the values of the vector of objectives for the first-period 
decision node. Note from Table 2-3 that a total of nine noninferior decisions are generated for action WA1. 
Similarly, there are five noninferior solutions for action DN1 (by comparison of all vectors for that action), 
and fourteen noninferior solutions after comparing all decisions for all actions (see Fig. 2-5). Fig. 2-6 
depicts the graph of all noninferior solutions. The decisionmaker, in interaction with the analyst, can usually 
identify a most-preferred policy from among the noninferior set. 

Example Problem for the Continuous Case 

Problem Definition 

The problem developed above for the discrete case is modified here to handle continuous loss functions 
and extreme random events. The set of objective functions is extended to include the conditional expected 
loss of lives and conditional expected cost in addition to the expected values. The main difference between 
the discrete and the continuous cases lies in calculating the damage vector for the terminal nodes, which 
can be determined using the expected value and/or the conditional expected value of extreme events. The 
subsequent computations are similar to those carried out for the discrete case. In addition, the assumption 
if )of the discrete case above is modified as: 

f) The parameters L and C are now defined as, respectively, the possible loss of lives and cost, given 
that no flood warning is issued; they are linear functions of the water flow W as defined in the 
technical section. 

The complete decision tree for this case is shown in Fig. 2-7. The loss functions L and C are used in 
calculating the unconditional expected-value denoted by f^*), and/or the conditional expected-value denoted 
by f 4 (*), which represents the risk of extreme events. Note that each of the risk measures the conditional 
expected value f 4 (*) and the unconditional expected value 5 f (•) is composed of two components (or 
expectation values) - cost and loss of lives. 


Summary of the Results 

The decision tree is simplified and solved for this continuous example as described in the technical 
section. Note that regardless of whether a watch (WA1) was issued or a do-nothing (DN2) action was 
followed at the first period, the same three possible actions are evaluated at the second period: evacuate, 
issue another flood watch, or do nothing. Depending on the actions taken at the first and second periods 
and the water flow level at the second period, different values of losses are generated for each terminal 
chance node. Recall that there are three equally probable underlying pdfs for the water flow for the first 
period. 


Table 2-4 summarizes the values of the unconditional and conditional expected loss vectors f 3 (*) 
and f 4 (*) for the decision arcs corresponding to the second period. Once these values are calculated, the 
noninferior decisions for each node are calculated by folding back the same way as in the discrete case. 
Table 2-5 yields the noninferior decisions for the second-period decision arcs. Averaging-out at the chance 
nodes for the first period follows the same procedure used in the discrete case. Consider, for example, 


26 






Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Table 2-3. Decisions for the First-Period Decision Node (Discrete Case) 


First- 

period 

decision 

Second-period decision 

Loss vector 

Higher 

Same 

Lower 

L 

C 

* EV1 

- 

- 

- 

0.0000 

5,177,940 

* WA1 

EV2 

EV2 

EV2 

0.2399 

4,578,391 

* WA1 

EV2 

EV2 

UA2 

0.2452 

3,689,511 

* UAl 

EV2 

EV2 

ON 2 

0.2506 

3,532,138 

UA1 

EV2 

UA2 

EV2 

0.2588 

3,925,796 

* VA1 

EV2 

UA2 

UA2 

0.2641 

3,036,916 

* UAl 

EV2 

UA2 

DN2 

0.2695 

2,879,543 

UAl 

EV2 

DN2 

EV2 

0.2777 

3,867,113 

UAl 

EV2 

DN2 

UA2 

0.2830 

2,978,233 

* UAl 

EV2 

DN2 

DN2 

0.2884 

2,820,860 

UAl 

UA2 

EV2 

EV2 

0.2776 

3,999,600 

UAl 

UA2 

EV2 

UA2 

0.2829 

3,110,720 

UAl 

UA2 

EV2 

DN2 

0.2883 

2,953,347 

UAl 

UA2 

UA2 

EV2 

0.2965 

3,347,005 

* UAl 

UA2 

UA2 

UA2 

0.3018 

2,458,125 

* UAl 

UA2 

UA2 

DN2 

0.3072 

2,300,752 

UAl 

UA2 

DN2 

EV2 

0.3154 

3,288,323 

UAl 

Ua2 

DN2 

UA2 

0.3207 

2,399,442 

* UAl 

UA2 

DN2 

ON 2 

0.3261 

2,242,070 

DN1 

EV2 

EV2 

EV2 

1.0136 

3,880,297 

DN1 

EV2 

EV2 

UA2 

1.0190 

2,991,417 

DN1 

EV2 

EV2 

DN2 

1.0566 

2,828,675 

DN1 

EV2 

UA2 

EV2 

1.0325 

3,227,701 

* DN1 

EV2 

UA2 

wa2 

1.0379 

2,338,821 

DN1 

EV2 

UA2 

DN2 

1.0756 

2,176,079 

DN1 

EV2 

DN2 

EV2 

1.1648 

3,150,115 

DN1 

EV2 

DN2 

UA2 

1.1702 

2,261,235 

DN1 

EV2 

DN2 

DN2 

1.2078 

2,098,493 

DN1 

UA2 

EV2 

EV2 

1.0513 

3,301,506 

DN1 

UA2 

EV2 

UA2 

1.0567 

2,412,626 

DN1 

UA2 

EV2 

DN2 

1.0943 

2,249,884 

DN1 

Wa2 

UA2 

EV2 

1.0702 

2,648,910 

★ DN1 

U|A2 

Ua2 

UA2 

1.0756 

1,760,030 

* DN1 

UA2 

UA2 

DN2 

1.1132 

1,597,288 

DN1 

UA2 

DN2 

EV2 

1.2025 

2,571,324 

DN1 

UA2 

DN2 

UA2 

1.2079 

1,682,444 

* DN1 

UA2 

DN2 

DN2 

1.2455 

1,519,702 

DN1 

DN2 

EV2 

EV2 

1.3153 

3,291,333 

DN1 

DN2 

EV2 

WA2 

1.3207 

2,402,453 

DN1 

DN2 

EV2 

DN2 

1.3583 

2,239,711 

DN1 

DN2 

UA2 

EV2 

1.3342 

2,638,737 

DN1 

DN2 

UA2 

UA2 

1.3396 

1,749,857 

ONI 

DN2 

UA2 

DN2 

1.3772 

1,587,115 

DN1 

DN2 

DN2 

EV2 

1.4665 

2,561,151 

DN1 

DN2 

ON 2 

UA2 

1.4719 

1,672,271 

* DN1 

DN2 

DN2 

DN2 

1.5095 

1,509,529 


* - noninferior decisions 


27 































Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 



Figure 2-5. Decision Tree for the First Stage (Discrete Case) 


28 
















Risk-Based Evaluation of 
Flood Wanting and Preparedness Systems 



Loss of Lives —** 


Figure 2-6. Pareto Optimal Frontier (Discrete Case) 


■ 


29 














Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 



\ (0 0L: 0.20 


10.2L; 0.80 
tO.1L; 0.40 

(0.2L! 0.50 

(0.3L; 0.90 

I (ML; 0.4 0 

(0.21: 0.50 

f0.3L; 0.90 

(O IL; 0.40 

(0.2L; 0.50 

10. 3L: 0.90 
MOL; 1.00 
i0.2L; 0.80 

(0.3L; 0.70 

U.OL; LOCI 

(Q.2L; 0.80 

10.3 L; 0.7Ci 

U.OL; LOO 

10.2L; 0.60 

(0.3L; 0,70 

U.OL; LOO 


Figure 2-7. Decision Tree for the Continuous Case 


30 




























Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Table 2-4. Loss Vectors for the Second-period Decision Arcs (Continuous Case) 


Node 

Arc 

f. 

5 ( * ) 

f 4 

(•) 

■L 

c 

L 

c 

D2 

• * EV2 

0.0798 

3,319,034 

3.5166 

17,066,293 


WA2 

0.1595 

898,792 

7.0332 

18,082,866 


• DN2 

0.2393 

717,826 

10.5497 

31,649,159 

D3 

• * EV2 

0.0312 

3,124,816 

1.9595 

10,838,047 


• * WA2 

0.062A 

656,020 

3.9190 

10,297,559 


• DN2 

0.0936 

280,835 

5.8785 

17,653,606 

DA 

• * EV2 

0.00A0 

3,016,172 

0.7598 

6,039,120 


• * UA2 

0.0081 

520,215 

1.5196 

4,298,901 


• DN2 

0.0121 

36,387 

2.2793 

6,838,021 

D5 

• * EV2 

0.1595 

3,478,550 

7.0332 

24,099,439 


Wa2 

0.2393 

1,058,309 

10.5497 

25,116,012 


• DN2 

0.7976 

797,584 

35.1657 

35,165,732 

D6 

• * EV2 

0.062A 

3,187,223 

3.9190 

14,757,071 


• * Wa2 

0.0936 

718,427 

5.8785 

14,216,583 


• DN2 

0.3120 

312,039 

19.5951 

19,595,118 

D7 

• * EV2 

0.0081 

3,024,258 

1.5196 

7,558,681 


• * WA2 

0.0121 

528,301 

2.2793 

5,818,461 


DN2 

0.0A0A 

40,430 

7.5978 

7,597,801 

C2 

F 

0.0886 

354,298 

4.4956 

17,982,472 

C3 

F 

0.AA29 

442,872 

22.4781 

22,478,090 


• noninferior decisions using f^(*) 

* noninferior decisions using f^(•) 


Table 2-5. Noninferior Decisions for the Second-period Decision Nodes (Continuous Case) 


Node 

Noninferior decisions 

f 5 C> 

f 4 ( *> 

D2 

EV2, WA2, DN2 

EV2 

D3 

EV2, UA2, DN2 

EV2, WA2 

D4 

EV2, UA2, DN2 

EV2, WA2 

D5 

EV2, WA2, DN2 

EV2 

D6 

EV2, WA2, DN2 

EV2, WA2 

D7 

EV2, UA2, DN2 

EV2, WA2 


31 
























Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


action WA1. There are 27 different combinations when using the expected value f,(«), and four different 
combinations when using f 4 («). Table 2-6 yields the values of the loss vectors for the first-period decision 
node using f 5 («), and Table 2-7 yields the values of the loss vectors using f 4 («). Note from Table 2-6 that 
for action WA1 there are a total of 10 noninferior decisions. Similarly for action DN1, there are 8 
noninferior solutions resulting from the comparison of all vectors for action DN1, and 6 noninferior 
solutions after comparison of all decisions for all actions using f,(«) (see Fig. 2-8). Figure 2-9 depicts the 
graph of all noninferior solutions using the traditional expected losses f,(«). 

Note from Table 2-7 that there is only one noninferior action when considering the conditional 
expected losses f 4 («). Thus, the action EV1 is the most conservative from the point of view of extreme 
events. When the decisionmaker considers the risk of extreme events, the potential loss of property 
overshadows the cost of the warning system. Thus, the two objectives — cost and loss of life — do not 
conflict in this example when looking only at the risk of extreme events. 


32 




Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Table 2-6. Decisions for the First-period Decision Node Using f 5 (Continuous Case) 


First- 

period 

decision 

Second-period decision 

Loss vector 

Higher 

Same 

Lower 

L 

c 

* EVl 

- 

- 

. 

0.0000 

5,088,574 

* WAl 

EV2 

EV2 

EV2 

0.0408 

3,781,716 

* WAl 

EV2 

EV2 

WA2 

0.0423 

2,888,912 

* WAl 

EV2 

EV2 

DN2 

0.0437 

2,715,847 

WAl 

EV2 

WA2 

EV2 

0.0492 

3,118,597 

* WAl 

EV2 

WA2 

WA2 

0.0507 

2,225,793 

* WAl 

EV2 

WA2 

DN2 

0.0521 

2,052,728 

WAl 

EV2 

DN2 

EV2 

0.0575 

3,017,822 

WAl 

EV2 

DN2 

WA2 

0.0590 

2,125,018 

* WAl 

EV2 

DN2 

DN2 

0.0604 

1,951,953 

WAl 

WA2 

EV2 

EV2 

0.0604 

3,184,884 

WAl 

WA2 

EV2 

WA2 

0.0619 

2,292,080 

WAl 

WA2 

EV2 

DN2 

0.0633 

2,119,015 

WAl 

WA2 

WA2 

EV2 

0.0688 

2,521,765 

* WAl 

WA2 

WA2 

WA2 

0.0703 

1,628,961 

* WAl 

WA2 

WA2 

DN2 

0.0717 

1,455,896 

WAl 

WA2 

DN2 

EV2 

0.0771 

2,420,990 

WAl 

WA2 

DN2 

WA2 

0.0786 

1,528,186 

* WAl 

WA2 

DN2 

DN2 

0.0800 

1,355,121 

WAl 

DN2 

EV2 

EV2 

0.0801 

3,140,258 

WAl 

DN2 

EV2 

WA2 

0.0816 

2,247,454 

WAl 

DN2 

EV2 

DN2 

0.0830 

2,074,389 

WAl 

DN2 

WA2 

EV2 

0.0885 

2,477,139 

WAl 

DN2 

WA2 

WA2 

0.0900 

1,584,335 

WAl 

DN2 

WA2 

DN2 

0.0914 

1,411,270 

WAl 

DN2 

DN2 

EV2 

0.0968 

2,376,364 

WAl 

DN2 

DN2 

WA2 

0.0983 

1,483,560 

* WAl 

DN2 

DN2 

DN2 

0.0997 

1,310,495 

DN1 

EV2 

EV2 

EV2 

0.1153 

2,851,964 

DN1 

EV2 

EV2 

WA2 

0.1167 

1,959,160 

DN1 

EV2 

EV2 

DN2 

0.1270 

1,784,649 

DN1 

EV2 

WA2 

EV2 

0.1236 

2,188,846 

* DN1 

EV2 

WA2 

WA2 

0.1250 

1,296,042 

* DN1 

EV2 

WA2 

DN2 

0.1353 

1,121,531 

DN1 

EV2 

DN2 

EV2 

0.1823 

2,079,690 

DN1 

EV2 

DN2 

WA2 

0.1837 

1,186,886 

DN1 

EV2 

DN2 

DN2 

0.1940 

1,012,375 

DN1 

WA2 

EV2 

EV2 

0.1350 

2,255,133 

DN1 

WA2 

EV2 

WA2 

0.1364 

1,362,329 

DN1 

WA2 

EV2 

DN2 

0.1467 

1,187,818 


33 



















Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


Table 2-6. (Continued) 


First 

Period 

Decision 

Second period decision 

Loss vector 

Higher 

Same 

Lower 

L 

c 

DNl 

WA2 

WA2 

EV2 

0.1433 

1,592,015 

* DNl 

WA2 

WA2 

Wa2 

0.1447 

699,211 

* DNl 

WA2 

WA2 

DN2 

0.1550 

524,700 

DNl 

WA2 

DN2 

EV2 

0.2020 

1,482,859 

DNl 

UA2 

DN2 

WA2 

0.2034 

590,055 

* DNl 

UA2 

DN2 

DN2 

0.2137 

415,544 

DNl 

DN2 

EV2 

EV2 

0.2727 

2,190,838 

DNl 

DN2 

EV2 

WA2 

0.2741 

1,298,034 

DNl 

DN2 

EV2 

DN2 

0.2844 

1,123,523 

DNl 

DN2 

WA2 

EV2 

0.2810 

1,527,720 

DNl 

DN2 

WA2 

UA2 

0.2824 

634,916 

DNl 

DN2 

WA2 

DN2 

0.2927 

460,405 

DNl 

DN2 

DN2 

EV2 

0.3397 

1,418,564 

DNl 

DN2 

DN2 

WA2 

0.3411 

525,760 

* DNl 

DN2 

DN2 

DN2 

0.3514 

351,249 


* noninferior decisions 


Table 2-7. Decisions for the First-period Decision Node Using f 4 (Continuous Case) 


First- 

Second-period decision 

Loss vector 

period 

decision 

Higher 

Same 

Lower 

L 

C 

* EV1 




0.0000 

9,495,618 

WAl 

EV2 

EV2 

EV2 

2.2367 

12,565,412 

WAl 

EV2 

EV2 

WA2 

2.5085 

11,942,936 

WAl 

EV2 

WA2 

EV2 

2.7630 

12,420,237 

WAl 

EV2 

WA2 

WA2 

3.0348 

11,797,761 

DNl 

EV2 

EV2 

EV2 

6.1876 

15,467,376 

DNl 

EV2 

EV2 

WA2 

6.4593 

14,844,900 

DNl 

EV2 

WA2 

EV2 

6.7140 

15,322,201 

DNl 

EV2 

WA2 

* . 

WA2 

Noninferioi 

6.9854 

: decisions 

14,699,725 


34 


























Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 



Figure 2-8. Decision Tree for the Second Stage Using f 5 (Continuous Case) 


35 
















Risk-Based Evaluation of 


Flood 


Warning and Preparedness Sy 


stems 



Figure 2-9. Pareto Optimal Frontier Using f 5 (Continuous Case) 


36 











Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Part 3 

Performance Characteristics of a 
Flood Warning System: Overview 

Introduction 

From the utilitarian point of view, rooted in the Bayesian principles of rationality, the ultimate 
measure of performance of a flood warning system is the ex ante economic value. From the engineering 
point of view, there remains the need for auxiliary measures that characterize, perhaps only partially, the 
performance of various components of a flood warning system. The purpose of such measures is to aid the 
engineer in the process of planning and design. 

One aspect of the performance of a flood warning system is its reliability. The following presents 
an overview of a model that outputs two measures of system reliability: 

• the relative operating characteristic (ROC), which shows a relationship among (i) the probability 
of detection, (ii) the probability of a false warning, and (iii) the expected lead time of a warning, 
and 

• the performance tradeoff characteristic (PTC), which shows a relationship among (i) the expected 
number of detections per year, (ii) the expected number of false warnings per year, and (iii) the 
expected lead time of a warning. 

Each characteristic, ROC and PTC, can be displayed graphically in the form of a family of curves. The 
displays offer an aid to engineering planning and design of flood warning systems. The concept and 
interpretation of these displays are illustrated with a case study of the flood warning system for Milton, 
Pennsylvania. 



Features of the Model 

System Model 

Structure 

The model is tailored to a class of local warning systems which can be conceptualized as a cascade 
coupling of three components, shown in Figure 3-1: monitor, forecaster, and decider. The operation of such 
a system is idealized as follows. 

Floods occur intermittently. For economic reasons, a flood data collection network, forecasting 
procedure, and emergency management do not operate continuously. Rather, their operation is triggered 
only when potential flood conditions are detected. To enable such detections, a system monitoring 
hydrometeorologic conditions operates continuously. When a set of predefined conditions is observed, the 


37 




Risk-Based Evaluation of 

Hood Warning and Preparedness Systems 



Figure 3-1. Functional Structure of a Flood Warning System 

monitor triggers operation of the forecast system. The flood data collection network is activated, and a 
forecast of the flood hydrograph is prepared. This forecast is supplied to the decision system - a flood 
preparedness organization, or a floodplain manager — who must then decide whether or not to issue a 
warning to the public. 

Assumptions 

Principal definitions and assumptions underlying our model of a flood warning system are as 

follows: 


1. A flood is the portion of a hydrograph above a flood stage, officially specified for a given river 
gauging station. 

2. If a flood forecast is prepared, it is issued at a well-defined instant, consistently for every flood. 
The performance of a warning system is evaluated based on this one forecast. 

3. The decision whether or not to issue a warning to the public is based on the forecasted flood 

crest. 


4. The flood plain is divided into elevation zones. A flood warning is issued for a zone. Thus, 
depending on the forecast, it may be optimal to issue a warning for a lower zone, but not for an upper zone. 
Consequently, the performance characteristics are defined for a zone. 

Mathematical models of the three system components are described below. 

Monitor 

An all-important design decision is the choice of a forecast trigger - an observable state that is 
likely to precede every flood and that, once observed, will trigger preparation of flood forecasts. Here are 
three examples of triggers: 

(river stage) exceeds (threshold) 

(rainfall intensity and duration) exceeds (threshold) 

(meteorologic situation) is among a set of {potential flood situations} 


38 





















Risk-Based Evaluation of 
Flood Wanting and Preparedness Systems 


A good monitor is particularly critical to local warning systems for flash floods in headwater areas 
with small watersheds and short concentration times. For instance, a flood developing rapidly during 
nighttime may occur undetected because of an equipment failure; a trigger may be false because the 
oncoming storm suddenly changes its track and bypasses the watershed. Consequently, the performance of 
the monitor may limit the performance of the total warning system, no matter how sophisticated its flood 
data collection network, forecasting procedure, and emergency management. 

Perfect diagnosticity of the monitor means that every trigger is followed by a flood; in other words, 
the monitor provides a perfect diagnosis of a flood situation. Perfect reliability of the monitor means that 
every flood is preceded by a trigger; in other words, the monitor never fails to signal the oncoming flood. 
Both monitor attributes, diagnosticity and reliability, are defined as probabilities and can be anywhere in 
the range from zero, or worst, to one, or perfect. 

Forecaster 

The objective of modeling is to obtain a stochastic characterization of floods and forecasts in the 
form requisite for decisionmaking and performance evaluation. Toward this end, a Bayesian processor of 
forecasts is formulated following the principles laid down in earlier works of Krzysztofowicz [1983a, 
1983b, 1985, 1987]. The inputs into the processor are a prior distribution describing natural flood events 
and a likelihood function describing the stochastic dependence between forecasted and actual flood events. 
The principal output from the processor is the posterior probability of flooding a given zone elevation, 
conditional on the forecast. This probability provides a basis for deciding the warning. In addition, the 
processor outputs several other probability distributions needed for system performance evaluation. The 
technical part outlines our approach to modeling the prior distribution and the likelihood functions. 

Decider 


When the trigger is observed and the forecast of the flood crest is prepared, the manager must 
decide whether or not to issue a flood warning for a zone of the floodplain. Thereafter the event takes 
place: either the zone is flooded or it is not. Each decision-event vector leads to outcomes whose 
undesirability (as they are mostly losses rather than gains) is evaluated in terms of a disutility function. The 
arguments of the disutility function are the actual flood crest and the lead time of the warning (to be defined 
later). 


The objective of decision analysis is to find the optimal warning rule. According to the Bayesian 
postulates of rationality, the optimal rule should minimize the posterior expected disutility of outcomes. For 
a statistical, as contrasted with the economic, evaluation of system performance, it is not necessary to find 
the exact form of the optimal decision rule. It suffices to know its general structure. In practical cases, the 
optimal warning rule is that a warning should be issued for a given zone whenever the posterior probability 
of flooding that zone, conditional on the particular forecast, exceeds a predetermined threshold value. 

Performance Measures 

Performance Probabilities 

The vector of binary indicators of the status of the trigger, warning, flood, and zone flood can take 
on nine values which define four performance states of the warning system, as shown in Figure 3-2: missed 


m 


39 





Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


flood (M), false warning (F), detection (D), and quiet. (Q). These states are observable in the sense that 
one could count their occurrences over a period of time. In the limit, this count would give rise to 
conditional probabilities of incorrect system performance, P(M) and P(F), and correct system performance, 
P(D) and P(Q). Since P(M) has a simple relationship to P(D), and, similarly, P(Q) to P(F), it suffices to 
find the probability of detection, P(D), and the probability of false warning, P(F). The objective of 
modeling is to express these probabilities in terms of parameters and functions which characterize the 
warning system. 

Relative Operating Characteristic 

Different disutility functions may result in different threshold values for warning. With each 
threshold value, there is associated a probability of detection P(D) and a probability of false warning P(F). 
A plot of P(D) versus P(F), obtained by varying the threshold, is called the relative operating characteristic 
(ROC). 


The ROC curve conveys the essential information about the tradeoffs that a given system offers 
between the probability of detection and the probability of false warning. However, the intuitive 
interpretation of these performance probabilities is not straightforward for they are conditional probabilities. 
Human intuition does not grasp easily such conditional events. Moreover, human cognition is generally not 
well trained in understanding and processing probabilities. Evidence of numerous and large biases in 
judgments involving probabilities is plentiful. 

Performance Tradeoff Characteristic 

In order to overcome the interpretive difficulties associated with the ROC, we propose to transform 
the probabilities of various states into the expected number of states per year. Given the expected number 
of floods per year the following quantities can readily be obtained: 

expected number of zone floods per year 

expected number of detections per year for a zone 

expected number of false warnings per year for a zone 

Once these expectation values are calculated, the ROC curve can be rescaled into a function relating the 
expected number of detections and the expected number of false warnings per year. This function will be 
called the performance tradeoff characteristic (PTC). 

Expected Lead Time 

The forecast time is the instant up to which hydrometeorologic observations for preparing the 
forecast are collected. The lead time of a warning for a given zone, conditional on the hypothesis that the 
zone will be flooded, is the time interval elapsed from the forecast time to the instant at which the flood 
waters reach the zone elevation. The lead time can be modeled as a random variable, and one can calculate 
its expected value. 


40 




Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


ZONE 

TRIGGER WARNING FLOOD FLOOD STATE 

T w 6 6 



Figure 3-2. Tree of Events Leading to One of the Four Performance States of a 

Flood Warning System 

(Missed Flood (M), False Warning (F), Detection (D), and Quiet (Q)) 


41 








Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


The designer of a warning system can affect the lead time indirectly, through the definitions of (i) 
the forecast trigger and (ii) the forecast time. Each of these specifications may affect the probability 
densities of forecasted flood crest and the lead time. Consequently, any change in the design specifications 
may simultaneously affect the following performance measures: probability of detection P(D), probability 
of false warning P(F), expected number of detections per year, expected number of false warnings per year 
NF, and the expected lead time. 

Closure 


The relative operating characteristic (ROC) and the performance tradeoff characteristic (PTC ) are 
a part of a general theory of flood warning systems that is being developed. A number of questions are still 
awaiting answers. Among them is the connection between these statistical measures of performance and the 
ex ante economic value of a warning system. Such a connection is well known within the classical detection 
paradigm, but it remains to be investigated whether or not it extends to a much more complex paradigm 
of a flood warning system. 

Applications of ROC and PTC concepts to other flood warning systems are also awaiting us. A 
number of applications would be desirable to systems with distinct hydrologic regimes, such as flash-flood 
streams and main-stem rivers, and distinct technologies, such as found in local warning systems and the 
forecast offices of the National Weather Service. Collectively, results of such case studies would offer 
useful guidance to engineers who plan and design flood warning systems. 


Case Study-Milton, Pennsylvania 

General Description 

Properties of the ROC and PTC curves, and their potential role as aids to design analysis, will be 
illustrated through a case study of the flood warning system for Milton, Pennsylvania. The town has a 
population of about 8000 and is located on the West Branch of the Susquehanna River in northeastern 
Pennsylvania. The data used in the study were collected by Krzysztofowicz and Davis [1983]. The source 
of the flood and forecast data is the U.S. National Weather Service, River Forecast Center at Harrisburg, 
Pennsylvania. The forecast data are from the period 1959-1975. Thus the case studies reported herein are 
representative of the system performance during that period. 

In all specifications of the parameters, the units of time are hours and the units of elevation are feet 
above the zero of the river gauge. The flood stage is at 19 ft, but almost all structures are located above the 
elevation of 22 ft. The densities in the model of the forecaster are assumed to follow the Gaussian law 
(normal probability distribution). 

Input Models and Parameters 

Record of floods. The record of floods from the period 1885-1975 contains 20 flood events. From 
this record, we estimated the expected number of floods per year and the prior density of flood crest. 


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Risk-Based Evaluation of 
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Models of Monitor and Forecaster. The parameters that must be estimated from the joint record of 
forecasted and actual floods are as follows: 

diagnosticity of the monitor, 

reliability of the monitor, 

two likelihood functions of forecasted flood crest, and 
-conditional on no flood occurring 
-conditional on the height of the flood crest 
expected lead time (for each zone elevation). 

Record of forecasts. The forecast verification reports for the period 1959-1975 contain a record of 
9 floods and 37 forecasts. The record does not contain information sufficient for the estimation of all 
parameters via statistical methods. Consequently, parameters of the monitor and parameters of the 
likelihood function that is conditional on no flood occurring had to be estimated subjectively based on a 
plausible interpretation and interpolation of the available information. On the other hand, the expected lead 
times and the parameters of the likelihood function that is conditional on flood crest were estimated from 
the data. 

System designs. The monitor is assumed to trigger the forecaster when the river stage exceeds a 
specified threshold. Three alternative system designs are analyzed, in which the forecast trigger is defined 
as follows: 

System SI: river stage exceeds 11 ft 

System S2: river stage exceeds 15 ft 

System S3: river stage exceeds 19 ft 

The likelihood function conditional on no flood is assumed to be the same for each system. The remaining 
parameters vary with the design. Table 3-1 lists estimates of the estimated expected lead times and the 
calculated expected number of zone floods for four zones of the floodplain. The respective elevations of 
the zones are given in the same table. 

Interpretation. Figure 3-3 shows the expected lead time plotted as a function of the elevation for 
each of the three systems. Clearly, when the threshold stage for triggering the forecaster is raised, the 
expected lead time is reduced uniformly for all elevations. Table 3-1 reveals further implications. When 
the lead time decreases, the diagnosticity of the monitor increases, since a higher threshold stage is always 
more diagnostic of the incoming flood. On the other hand, when the lead time decreases, the reliability also 
decreases. This is so because the observations of the river are made in 6-hour intervals, and it is possible 
for a rapidly rising river to exceed both the threshold stage and the flood stage within the 6-hour interval. 
In such an instance, flooding occurs prior to the preparation of a forecast. The likelihood of such an event 
increases as the threshold stage is raised closer to the flood stage; hence the reliability decreases. 


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ZONE ELEVATION y [ft] 


Figure 3-3. Expected Lead Time of a Flood Warning Versus the Elevation of 
the Floodplain for Three Warning Systems in Milton, Pennsylvania 


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Table 3-1. Parameters of Three Alternative Designs of a Flood Warning System 

for Milton, Pennsylvania 


Monitor Likelihood Function Forecast 


oysL&m 

Design 

Diagnosticity 

If 

Reliability 

P 

Slope 

a 

Intercept 

b 

St. Dev. 

0 

Characteristic 

FSC 

SI 

0.80 

1.00 

0.44 

10.65 

3.06 

6.95 

S2 

0.90 

0.89 

0.45 

12.10 

1.90 

4.22 

S3 

1.00 

0.83 

0.64 

8.48 

2.21 

3.45 


Table 3-2. Expected Number of Zone Floods and Expected Lead Times of Flood Warnings 

for Milton, Pennsylvania 


Zone 

Elevation 
y £ ft ] 

Expected 
Number of 
Zone Floods 

n 

Expected Lead Time 

LT [hrs] 

System 

SI 

System 

S2 

System 

S3 

19 

47.1 

9 

5 

-3 

22 

38.4 

15 

11 

4 

25 

26.0 

21 

17 

11 

28 

13.7 

27 

24 

18 

*The expected numbers are 

for the per 

iod of 100 

years. 


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Risk-Based Evaluation of 

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When the expected lead time decreases, one also anticipates an increase in the quality of the flood 
crest forecasts. Table 3-1 reveals that the parameters of the likelihood function change their values with a 
change in lead time. But do these changes imply anything about the forecast quality? The answer to this 
question may be obtained via the forecast sufficiency characteristic (FSC). This measure is sufficient for 
comparing any two forecasters who produce forecasts and enables us to order forecasts in terms of their 
economic values. The FSCs calculated in the last column of Table 3-1, along with the expected lead times 
for the three systems in Table 3-2, confirm our hypothesis: when the lead time decreases, the quality of the 
flood crest forecasts increases. 

Properties of the ROC and the PTC 

The ROC and PTC curves for design SI are displayed in Figures 3-4 and 3-5. We shall highlight 
some general properties of these curves. 

1. For a fixed zone elevation, the associated ROC is a concave function specifying a unique 
relationship between the probability of false warning, P(F), and the probability of detection, P(D). 
Probability P(F) may vary from 0 to 1, but probability P(D) is bounded from above by the reliability of the 
monitor p, which for design SI happened to be 1.0. For a fixed zone elevation and a prior distribution of 
the flood crest, the shape of the ROC curve depends solely upon the design specifications for the monitor 
(via diagnosticity y and reliability p) and the design specifications for the forecaster (via the likelihood 
functions of flood crest forecast: one conditional on the flood crest, the other conditional on no flood 
occurring). 

2. By mapping each point from the ROC in Figure 3-4 through relations (11)-(12) given in the 
technical part, we obtain the PTC shown in Figure 3-5. The PTC is also a concave function, increasing 
from the origin, which corresponds to P(F) = P(D) = 0, to a point which corresponds to P(F) = 1 and 
P(D) = p. The expected number of detections per year, ND, never exceeds the expected number of zone 
floods n. On the other hand, the expected number of false warnings per year, NF, may exceed n. The PTC 
for zone elevations y = 25 and y = 28 do just that. 

Performance Differences Between Zones 

1. The ROC curves for different elevation zones have generally similar shapes and cross each other. 
In other words, when the performance of a warning system is measured in terms of the probability of false 
warning P(F) and the probability of detection P(D), all zones seem to be served equally well. However, the 
third performance measure - the expected lead time of the warning - illuminates the differences between 
the low-lying zones and the high-lying zones: LT = 9 hours for y = 19 and LT = 27 hours for y = 28, 
a threefold difference. 

2. The PTC curves are quite dissimilar, underscoring the fact that they convey different information 
than the ROC curves do. There are two main distinctions between the zones. First, there is the obvious 
distinction resulting from the elevations difference: the expected number of floods n in 100 years is 47.1 
for y = 19 and only 13.7 for y = 28. Second, there is a remarkable difference in terms of the expected 
number of false warnings NF associated with the maximum expected number of detections ND = n. This 
NF is equal to 19.2 for y = 19, and it increases to 52.5 for y = 28. In other words, to reach the upper 
limit of expected detections for the higher zone, one must accept a rate of false warnings NF = 52.5, which 
is 3.8 times higher than the rate of floodings n = 13.7. 


46 




PROBABILITY OF DETECTION P(D) 


Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 



PROBABILITY OF FALSE WARNING P(F) 


Figure 3-4. Relative Operating Characteristics (ROC) of Warning System SI 
for Four Zone Elevations, y, in Milton, Pennsylvania. 

(Symbol LT Denotes the Expected Lead Time of a Flood Warning) 


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3. To place these results in proper perspective, one has only to realize that floods reaching zone 
y = 28 are more extreme and rare than floods reaching only zone y = 19. The PTC curves in Figure 3-5 
inform us that a high detection rate for rare events comes at the price of a high rate of false warnings. This 
appears to be an inescapable tradeoff. 

Operating Points 

1. A point on the ROC, or PTC, is called an operating point. In Figure 3-4, we fixed an operating 
point for each zone such that for all zones the probability of false warning P(F) = 0.25. The probability 
of detection P(D) is different for each zone, but the differences are small. Table 3-3 lists the exact 
coordinates of these operating points on PTC. Figure 3-5 depicts these points, each of which has distinct 
NF and ND coordinates. 

Table 3-3. Coordinates of Operating Points on the ROC and PTC Curves That Give 
the Same Probability of False Warning P(F) for Each Zone Elevation; 

System Design SI for Milton, Pennsylvania 


Operating 

Point 

Zone 

Elevation 

yfft] 

Probability of 

Expected Number of 

Expected 
Lead Time 
LT [hrs] 

Detection 

P(D) 

False W. 
P(F) 

Detections 

ND 

False W. 
NF 

A 

19 

0.75 

0.25 

35.1 

4.8 

9 

B 

22 

0.71 

0.25 

27.3 

6.9 

15 

C 

25 

0.69 

0.25 

18.1 

10.0 

21 

D 

28 

0.72 

0.25 

9.9 

13.3 

27 


*The expected numbers are for the period of 100 years. 


2. In general, the mapping between the operating points of ROCs and PTCs is one-to-one, with the 
following properties: (i) The operating points that have the same P(D) coordinate on the ROC have also the 
same ND coordinate on the PTC. (ii) TTie operating points that have the same P(F) coordinate on the ROC 
may have different NF coordinates on the PTC. We shall say that the mapping between the ROC and PTC 
is nonorthogonal. 


3. The nonorthogonality of the mapping between the ROC and PTC should be taken as a caution: 
judgmental analysis of tradeoffs on the ROC, or PTC, is not a simple cognitive task. We recommend using 
the PTC as the primary aid to planning and design because the expected numbers ND and NF are easier 
to interpret and understand than the probabilities P(D) and P(F), which are conditional probabilities. 


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NUMBER OF FALSE WARNINGS NF 


Figure 3-5. Performance Tradeoff Characteristics (PTC) of Warning System SI for 
Four Zone Elevations, y, in Milton, Pennsylvania. 

(Symbol N Denotes the Expected Number of Zone Floods. All numbers, NF, ND, 
and n are for the Period of 100 Years.) 


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4. With each operating point on the PTC, or ROC, there is associated a unique threshold q’ in the 
warning rule. Thus, a specification of the operating point is equivalent to a specification of the rule for 
deciding warnings. To specify an operating point on the PTC for a given zone, one should consider a 
tradeoff between the expected number of detections ND and the expected number of false warnings NF. 
This tradeoff should encapsulate one's preferences for outcomes of all possible decision-event vectors for 
this particular zone. It follows that it would be irrational to fix the operating point based solely on a 
displayed PTC or ROC, without an in-depth analysis of all socioeconomic outcomes of every decision-event 
vector. That is why the PTC and ROC curves should be viewed only as aids to the planning and design 
process, rather than as a means of specifying the warning rule. The optimal warning rule should be found 
by minimizing the expected disutility of outcomes resulting from all possible decision-event vectors. 

Performance Tradeoffs 

From a purely statistical point of view, which ignores the economic and social decision criteria, the 
engineer could consider the design process as an optimization problem with three criteria: maximize the 
expected number of detections ND, minimize the expected number of false warnings NF, and maximize 
the lead time LT. The ideal solution is an operating point at which: 

-the expected number of detections ND is equal to the expected number of zone floods n, 

-the expected number of false warnings NF is equal to zero, and 

-the expected lead time LT is equal to infinity. 

In the absence of the ideal solution, tradeoffs must be made. 

The kinds of tradeoffs that one may encounter are illustrated for the three alternative system 
designs, SI, S2, and S3. The ROC and PTC curves of these systems are compared in Figures 3-6 and 3-7 
for zone elevation y = 22 and in Figures 3-8 and 3-9 for zone elevation y = 28. An immediate observation 
is that designs SI and S2 offer distinct performance characteristics. On the other hand, designs S2 and S3 
have similar ROC and PTC curves over a range of operating points, while over the remaining range S2 
outperforms S3. Together with the fact that S3 offers much shorter expected lead times LT than S2 does, 
it is unlikely that decisionmakers would prefer S3 over S2. This example illustrates then a screening analysis 
that may be performed on a large set of alternative designs before a few are selected for a detailed analysis 
of tradeoffs. 

The ensuing discussion highlights the nature of performance tradeoffs that the PTC allows one to 
analyze between designs SI and S2. The discussion concentrates on three operating points, labeled A, B, 
C, in Figure 3-7. Their coordinates (ND, NF, LT) are listed in Table 3-4. 

1. A good way to start the analysis is to fix the expected number of false warnings NF at a level 
that appears acceptable, at least initially, say NF = 5.0, which means that one would be willing to accept 
5.0 false warnings in 100 years, on the average. At this level of NF, design SI ensures the expected 
detection of ND = 23.8 floods out of the expected n = 38.4 floods in 100 years. The expected lead time 
of a warning for each detected flood is LT = 15 hours. The difference, n - ND = 38.4 - 23.8 = 14.6, is 


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Risk-Based Evaluation of 
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PROBABILITY OF FALSE WARNING P(F) 


Figure 3-6. Relative Operating Characteristics (ROC) of Three Warning Systems, SI, S2, 
and S3 for Zone Elevation y = 22 ft in Milton, Pennsylvania. 

(Symbol p denotes the reliability of the monitor) 


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NUMBER OF DETECTIONS ND 


Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 



Figure 3-7. Performance Tradeoff Characteristics (PTC) of Three Warning Systems, 
SI, S2, and S3, for Zone Elevation y = 22 ft in Milton, Pennsylvania. 


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p 



PROBABILITY OF FALSE WARNING P(F) 


Figure 3-8. Relative Operating Characteristics (ROC) of Three Warning Systems, 
SI, S2, and S3, for Zone Elevation y = 28 ft in Milton, Pennsylvania. 
(Symbol p denotes the reliability of the monitor) 


S3 










Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 



NUMBER OF FALSE WARNINGS NF 


Figure 3-9. Performance Tradeoff Characteristics (PTC) of Three Warning Systems, SI, S2, 
and S3, for Zone Elevation y = 22 ft in Milton, Pennsylvania 


Table 3-4. Coordinates of Three Alternative Points on the ROC and PTC Curves 

for Zone Elevation y = 22 ft 



System 

Design 

Operating 

Point 

Probability of 

Expected Number* of 

Expected 

Detection 

P(D) 

False W. 
P(F) 

Detections 

ND 

False W. 
NF 

Lead Time 

LT [hrs] 

SI 

A 

0.62 

0.18 

23.8 

5.0 

15 

SI 

B 

0.75 

0.29 

28.9 

8.1 

15 

S2 

C 

0.75 

0.27 

28.9 

5.0 

11 

*The expected numbers are for the period of 

100 years. 




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the expected number of floods in 100 years that will arrive undetected, and thus will not be preceded by 
a warning to the public. 

2. At the same level of NF = 5.0, design S2 ensures the expected detection of 28.9 floods in 100 
years, with the expected lead time of a warning equal to 11 hours; the expected number of missed floods 
in 100 years is 38.4 - 28.9 = 9.5. Thus, when comparing the operating points A and B, the following 
tradeoff should be considered: is it preferable or not to reduce LT from 15 to 11 hours in order to increase 
ND from 23.8 to 28.9 (or, equivalently to reduce the expected number of missed floods from 14.6 to 9.5)? 

3. A similar analysis of tradeoffs may be carried out for a fixed expected number of detections ND, 
say 28.9 in 100 years. At this level of ND, the number of false warnings expected in 100 years is 5.0 for 
design S2 and 8.1 for design SI; the accompanying expected lead times of a warning are, respectively, 11 
and 15 hours. Thus, when comparing the operating points B and C, the following tradeoff should be 
considered: is it preferable or not to reduce LT from 15 to 11 hours in order to decrease NF from 8.1 to 
5.0? 


4. The right endpoints of the PTC curves indicate that, given the present specifications for the 
monitor, design SI can detect all n = 38.4 floods expected in 100 years. However, design S2 has an upper 
limit of 34.2 expected detections in 100 years; thus, the minimum expected number of missed floods in 100 
years is 38.4 - 34.2 = 4.2. The upper limit of ND is achieved by each design at a different level of the 
expected number of false warnings NF, which is 27.8 for design SI, and 18.2 for design S2. 


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Risk-Based Evaluation of 
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Part 4 

Selection of Optimal Flood 
Warning Threshold: 

Overview 

Introduction 

Flood control can be provided by either structural or nonstructural measures or a combination of 
both. Structural flood control measures, such as an increase in dam height, affect the flood-frequency 
relationship. Nonstructural measures, such as a flood warning system, do not have an impact on the flood- 
frequency relationship; however, they modify the flood-damage relationship. 

In this chapter, flood warning systems are studied in a two-level hierarchical system framework. 
The interactions between the forecast subsystem and the response subsystem are investigated. Emphasis is 
placed on exploring the impact of the current selected flood warning threshold on the future response 
fraction of a flood warning. The probabilistic evaluation of a forecast system coupled with a stochastic 
dynamic model of the evolvement of the response fraction in a community reveals that the desire for high 
present flood-loss reduction must be balanced with the possibility of high future flood loss. Multiobjective 
dynamic programming is used to select the optimal flood warning threshold. The proposed methodology 
is applied to case studies in Milton, Eldred, and Connellsville, Pennsylvania. 



Features of the Model 

In general, the overall flood warning system can be viewed as and modeled in a two-level 
hierarchical system framework as is depicted in Figure 4-1. There are two subsystems at the lower level. 
One is the forecasting subsystem, which issues a flood forecast based on hydrological and climatic 
information. The other is the response subsystem, which includes decisionmaking and action implementation 
of a community in response to flood warning. At the upper level, it is assumed that a regional agency exists 
whose functions are to set a warning threshold, disseminate a flood warning to the community, provide 
transportation during the evacuation process, and collect statistical data of the warning system. 

Performance Measures of a Warning System 

Define a random variable which represents the actual flood crest and another random variable which 
represents the forecasted flood crest. If the prior probability density function of the flood crest is known 
and the conditional probability density function of forecasted crest, given the actual crest, is known, then 
the posterior probability density function of the actual crest, given the forecasted crest, can be 
straightforwardly obtained. 

In this pan both the prior distribution of the flood crest and the likelihood function are assumed to 
be of normal distributions, employing the so-called normal-linear model. With the assumption of the 


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Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


normal-linear model, the probability distributions used throughout the analysis are of particularly simple 
forms. 



Figure 4-1. Multilevel Structure of Flood Warning Systems 

The flood warning threshold is defined by the fact that a warning is issued when the forecasted crest 
exceeds this threshold. A flood warning will be issued only when the forecasted flood crest exceeds this 
preassigned threshold level. For a given physical forecast system, the performance of the system can be 
evaluated by four probabilistic measures. There exist four possible outcomes that follow a flood warning 
decision: a correct warning, a false warning, a missed warning, and a correct quiet (the decision not to issue 
a warning). A correct warning is a warning followed by a flood. A false warning is a warning not followed 
by a flood. The probability of a false warning is also easily obtained. A missed forecast is a flood event 
which is not preceded by a warning. A correct quiet is the case of no warning and no flood. The 
probabilities of these four outcomes can be obtained from the assumptions concerning the forms of the 
probability distributions and likelihood functions. These four probabilistic measures are related to each 
other. Knowing one of them and the prior flood probability and the probability density function of the 
forecasted crest, the other three can be calculated. 

There are two types of prediction errors of a forecast system -- Type I and Type II errors. Type 
I errors are those of missed predictions. Type II errors are those of false alerts. Clearly, the value of the 
selected warning threshold plays a key role in determining the probabilities of Type I and Type II errors. 
If the threshold is set lower, the forecast will have a lower probability value of a Type I error and a higher 
probability value of a Type II error. If the threshold is set higher, the forecast will have a higher probability 
value of a Type I error and a lower probability value of a Type II error. 

Type I and Type II errors have different impacts on flood-loss reduction. A Type I error will result 
in an immediate flood loss. Thus, it has mainly a short-term impact. On the other hand, a Type II error will 
reduce the credibility of the forecast system. This cry-wolf consequence has mainly a long-term impact. 


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Such errors do not cause a flood loss at the present stage, but will discourage the response to flood 
warnings for future flood events. The present fraction of people who respond to a flood warning is certainly 
an indispensable factor in constructing the flood-loss function for a community. It thus affects the selection 
of the flood warning threshold. On the other hand, the response fraction fluctuates as time passes, based 
on the past performance of the warning system. The coupling between successive flood events is carried 
by dynamic evolvement of the fraction of people who respond to a flood warning. The past performance 
of a flood warning system affects the present fraction of people who respond to the warning system. 
Therefore, it is necessary to investigate the operation of flood warning systems in a dynamic framework. 
Figure 4-2 shows the connection between the flood warning threshold and the response fraction of the 
community. 

A Model of the Response System 

In general, the response of a community to a flood warning system is affected by people's 
experience of flooding and their subjective evaluation of the past performance of the forecasting system. 

The general interaction between a forecasting system and a response system can be described from 
the following considerations. The effectiveness of a forecasting system can be judged from its performance 
measures of Type I and Type II errors. The response of a community to a flood warning can be described 
by a state variable, that is, the fraction of people in the community who respond to a call for evacuation 
when warned. If a past flood event has been predicted, then confidence in the flood forecasting system will 
increase, and thus, future rates of response will also increase. On the other hand, a cry-wolf (Type II error) 
event will decrease confidence in the flood forecasting system, thereby decreasing future rates of response. 
People tend to have decreased confidence in a flood warning system when they have experienced a missed 
warning. However, the experience of flooding will increase people's alertness to the possibilities of future 
floods. For simplicity, it is reasonable to assume that the response fraction will remain unchanged after a 
missed warning has been experienced. It is also assumed that a correct quiet does not change the response 
fraction in the future. In view of the above discussion, the response fraction evolves dynamically, governed 
by some underlying law of transition. 

The response fraction in the community is described here as a controlled stochastic process in which 
the selected value of the warning threshold controls the transition probabilities. Knowing the present value 
of the response fraction, three possible transitions exist with given probabilities. The actual transition 
depends on the real outcome associated with the present warning decision. This stochastic system can be 
controlled in the sense of the expected value. 

Note here that the feedback loop that encompasses the forecast and the response subsystems is 
closed only when the next flood event occurs. The present performance of a forecast system does not affect 
the response fraction at the present flood event, but it does affect the response fraction at the next flood 
event. 

Multiobjective Multistage Optimization Model 

A key aspect of flood warning systems is that the selection of the flood warning threshold cannot 
be viewed in isolation at each single flood event since the decisionmaker must balance the desire for high 
present flood-loss reduction with the possibility of high future flood loss. A multiobjective multistage 
optimization model is proposed in this part for finding the best value for the flood warning threshold of a 


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Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


flood warning system. Evaluating the tradeoff between short- and long-term effects yields to an acceptable 
balance between the expected loss reduction at the current stage and the fraction of people who respond at 
the next flood event. 


Response 

Fraction 


Decision Logic 


Flood Warning 


Response 

T+l 

(Upper Level) 

★ 

S T 

System 

t’ 

Sys tern 



Warning Probabilistic 

Threshold Performance 


Measures 


Figure 4-2. Interaction Between Forecast and Response Subsystems 

At each flood event, the maximization of two noncommensurate and/or conflicting objective 
functions is considered. At a particular flood event, one objective is to maximize the expected property-loss 
reduction; the second objective is to increase the system's credibility by reducing the cry-wolf effect (i.e., 
increasing the fraction of people who respond in the future). 

There are four possible, alternative losses associated with the warning decision: the expected loss 
when no warning is given and no flood occurs (assumed to be zero), the expected community property loss 
with no warning followed by a flood, the cost of an unneeded evacuation in the community (a warning and 
no flood), and the expected community property loss with a warning (warning followed by a flood). It is 
evident that the response fraction affects the cost of evacuation and the property loss with a warning. The 
technical section describes how to calculate the following measures of performance: (1) the expected 
property loss with a warning system operating at a selected threshold rule, (2) the expected property loss 
without a warning system, and (3) the expected property loss reduction realized from installing a warning 
system. All these loss functions, which are specific to an individual flood event, are described in detail in 
the technical section. 

To construct a reasonable loss function, the concept of category-unit loss functions proposed by 
Krzysztofowicz and Davis [1983d] is adopted. The main modification is that the concept of the response 
degree of an individual, which was orig inall y used in Krzysztofowicz and Davis [1983], is used in this study 
to represent the fraction of people who respond to the warning. 

The N-stage multiobjective optimization problem of flood warning systems is then formulated to 
maximize the sum of the expected property-loss reductions of all flood events over the time horizon under 
consideration, and to maximize the forecast system's credibility, which is implicitly expressed by the 
expected fraction of people who respond to the warning beyond the time horizon under consideration. A 
third objective of ma ximizin g the expected life-loss reductions of all flood events over the time horizon 
under consideration can also be included in this problem formulation when data are available. 

Solving a multiobjective multistage optimization problem yields the set of noninferior solutions. A 
solution is noninferior if there is no other solution which improves on a single objective without seeing 


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Risk-Based Evaluation of 
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losses in one or more other objectives. In this example, the selection of an optimal warning threshold from 
among the noninferior alternatives involves tradeoffs between the probabilities of Type I and Type II errors. 
The multiobjective optimization can be solved by the weighting method and dynamic programming. 


Case Studies 

In the following case studies, historical data records obtained from the National Weather Service 
were used to estimate moments of the actual and forecasted crests; these estimates were next employed as 
parameter values in the normal-linear model of a forecast system. It must be stressed that this approach, 
while convenient analytically, offers at best an approximate representation of uncertainties in flood crests 
and their forecasts. Rather, the results presented in this report should be treated as hypothetical examples 
having some, but not all, realistic features. 


Application to Milton, Pennsylvania 

System design S2 for Milton, Pennsylvania, described in this report in Part 3, Performance 
Characteristics of a Flood Warning System, is selected as the basis for the study in this subsection. In this 
case, the flood crest is of a normal distribution and the conditional probability density function of the 
forecasted flood crest, given the actual crest, is of a normal-linear form. It can be shown that (I) the 
marginal probability density function of the forecast is of a normal distribution and (2) the posterior 
distribution density function of the actual crest, given the forecasted crest, is of the normal-linear form. The 
probability of flooding given a particular forecasted crest is obtained by means of these distributions. 

The four probabilistic measures of a forecasting system can be calculated for a given warning 
threshold and zone elevation. Table 4-1 shows those measures for two values of zone elevation and various 
values of warning threshold. A tradeoff between Type I and E errors can be clearly seen from Fig. 4-3. 
Different values of warning threshold are associated with different values of the probabilistic measures: 
probabilities of correct warning, correct quiet, false warning, and missed warning. They thus yield different 
impacts on the response fraction at the subsequent stage. 

As in the other case studies, reasonable assumptions concerning the loss-function parameters are 
made that, along with the probabilities discussed above, enable the calculation of the expected flood-loss 
reduction. 

Table 4-2 gives the calculated values of the expected flood-loss reduction for various response 
fractions and preselected warning thresholds where zone elevation is equal to 19 or 22 feet. The relationship 
between the expected flood-loss reduction and the warning threshold is also depicted in Fig. 4-4 for various 
response fractions and elevation levels. We consider a five-stage problem with the initial responding fraction 
equal to 0.7; that is, 70 percent of the population will respond to a warning initially. Two values of the 
elevation are considered, 19 and 22 feet. 


61 







Risk-Based Evaluation of 

flood Warning and Preparedness Systems 


Table 4-1. Probabalistic Measures of the Warning System 
y - 19 


* 

* 

* 

* 

* 

s 

p u (s .y) 

p io (s ■*> 

p oi (s •*> 

p oo (s •*> 

15.0 

0.8885870 

0.1094329 

0.0000862 

0.0018939 

15.5 

0.8884591 

0.1081553 

0.0002140 

0.0031715 

16.0 

0.8881721 

0.1061974 

0.0005011 

0.0051294 

16.5 

0.8875671 

0.1033221 

0.0011061 

0.0080047 

17.0 

0.8863676 

0.0992858 

0.0023056 

0.0120411 

17.5 

0.8841263 

0.0938829 

0.0045469 

0.0174439 

18.0 

0.8801783 

0.0870001 

0.0084949 

0.0243263 

18.5 

0.8736137 

0.0786737 

0.0150595 

0.0326531 

19.0 

0.8632919 

0.0691255 

0.0253813 

0.0422013 

19.5 

0.8479231 

0.0587649 

0.0407501 

0.0525619 


y - 22 


* 

* 

* 

* 

* 

s 

p u <® .y) 

P 10 (s ’ y> 

p oi (s • 

p oo^ s 

19.0 

0.7171233 

0.2152941 

0.0077255 

0.0598570 

19.5 

0.7108782 

0.1958098 

0.0139707 

0.0793414 

20.0 

0.7009019 

0.1734535 

0.0239469 

0.1016977 

20.5 

0.6858719 

0.1490539 

0.0389770 

0.1260972 

21.0 

0.6644858 

0.1237766 

0.0603631 

0.1513745 

21.5 

0.6357059 

0.0989638 

0.0891429 

0.1761873 

22.0 

0.5990129 

0.0759250 

0.1258360 

0.1992262 

22.5 

0.5546101 

0.0557205 

0.1702388 

0.2194306 

23.0 

0.5035065 

0.0390087 

0.2213424 

0.2361424 

23.5 

0.4474492 

0.0259864 

0.2773997 

0.2491647 


62 



































Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 




63 










Risk-Based Evaluation of 

Flood 'Warning and Preparedness Systems 


Table 4-2. Expected Flood-loss Reduction 


y - 19 


* a 

s 

0.50 

0.60 

0.70 

0.80 

16.5 

6.013321 

7.215985 

8.418648 

9.621314 

17.0 

6.014395 

7.217274 

8.420152 

9.623033 

17.5 

6.011573 

7.213888 

8.416201 

9.618517 

18.0 

5.998908 

7.198690 

8.398471 

9.598253 

18.4 

5.975090 

7.170107 

8.365125 

9.560144 


y “ 22 


* Q 

S 

0.50 

0.60 

0.70 

0.80 

19.0 

3.592137 

4.310565 

5.028992 

5.747422 

19.5 

3.595634 

4.314761 

5.033888 

5.753016 

20.0 

3.574387 

4.289264 

5.004142 

5.719021 

20.5 

3.511868 

4.214242 

4.916617 

5.618992 

20.9 

3.420732 

4.104879 

4.789026 

5.473174 


64 

























Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 



5(a) 



5(b) 


Figure 4-4. Relationship Between the Expected Flood-loss Reduction and 
the Warning Threshold for Various Response Fraction and Elevation Levels 


65 









Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 



OJ C\J C\J 

5(c) 



CM CM CM 


5(d) 


Figure 4-4. (continued) 


66 








Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


The overall problem can now be posed as follows: choose the optimal warning threshold to 
maximize the sum of flood loss reductions in all five stages and the response fraction of the community after 
five stages. The optimization problem can be solved by the weighting method and dynamic programming. 

(1) the lower the weighting coefficient associated with the first objective (loss reduction), the higher 
the value of the flood warning threshold will be set to avoid possible high Type II errors; 

(2) in order to select a decision that maximizes the sum of flood-loss reductions, the flood warning 
threshold is set higher at the earlier stage than at the later stage (with respect to the same value of the 
response fraction) in order to reduce the probability of high loss at the later stages; and 

(3) the higher the present response fraction, the more cautious the selection of the threshold is. That 
means that a higher value of threshold is set for a higher value of the present response fraction in order to 
avoid losing a larger number of the response population. We should note here that the third conclusion may 
be model-specific. 

Figures 4-5, 4-6, and 4-7 show the optimal flood warning threshold as functions of the weighting 
coefficient, the stage, and the response fraction, respectively. 


Application to Eldred, Pennsylvania 

Location 

Eldred is a small community situated on the upper Allegheny River in northern Pennsylvania, about 
three miles south of the state border with New York. The river gauge has a datum at 1417 feet and closes 
a drainage area of 50 square miles of mountainous terrain with its highest ranges towering at 2500 feet. The 
river flow at Eldred is essentially unimpaired natural runoff. 

Data Records and Parameter Estimation 

Historical flood and forecast data were retrieved from the archives of the National Weather Service 
Forecast Office in Pittsburgh. The prior distribution of flood crests was estimated from a record of floods 
spanning 1942-1989. During these 48 years, 36 flood crests exceeded the gauge height of 11 feet (3 floods 
in every 4 years, on the average), and 14 had crests above the official stage of 17 feet (about one flood in 
every 3.5 years, on the average). The highest flood on record occurred in June 1972 and reached 29 feet. 

The likelihood functions were estimated from a historical joint record of forecasted and actual flood 
crests. This record contained 12 floods that occurred in the period 1984-1988. 

Results of Case Study 

From the historical data, the flood crest is fitted by a normal distribution, and the conditional 
probability density function of the forecasted flood crest, given the actual crest, is also fitted by a normal 
distribution (the normal-linear model). It can be shown that 

(a) the marginal probability density function of the forecasts is also of a normal distribution, and 


67 





Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


s* 




stage 4 



Figure 4-5. Relationship Between the Optimal Warning Threshold and the 
Weighting Coefficient for a = 0.55 at Stage 4 






Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


S* 



stage 


Figure 4-6. Relationship Between the Optimal Warning Threshold and the 
Stages for a = 0.70 and 0 = 0.1 


69 








Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 



a 


oooooooo 


Figure 4-7. Relationship Between the Optimal Warning Threshold and the 
Response Fraction for 0 = 0.02 at Stage 3 


70 











Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


(b) the posterior distribution density function of actual flood crest given the forecasted crest is of 
the normal-linear form. 

The four probabilistic measures of the forecasting system can be calculated for a given warning 
threshold and zone elevation. Different values of warning threshold are associated with different values of 
the four probabilistic measures: probabilities of correct warning, correct quiet, false warning, and missed 
warning. They thus yield different impacts on the response fraction at the subsequent flood stages. The 
flood-loss information is not available in the case study, and hence reasonable assumptions are made about 
the parameters of the loss functions. 


Application to Connellsville, Pennsylvania 

Location 

Connellsville, a town in southwestern Pennsylvania, embraces the banks of the Youghiogheny 
River-a tributary of the Monongahela River. The river gauge has a datum at 860 feet and closes a drainage 
area of 1326 square miles.The terrain varies from hilly to mountainous, especially in the eastern part of the 
basin where Mt. Davis reaches 3213 feet-the highest point in Pennsylvania. 

Reservoirs 

The river flow in Connellsville is partly regulated by storage reservoirs. The Deep Creek Reservoir, 
completed in January 1925, is used for hydroelectric power generation. It is owned an operated by the 
Pennsylvania Electric Company. The reservoir has a capacity of 93,000 acre-feet and closes a drainage area 
of 65 square miles, or about 5% of the total basin. Thus its influence on flood flows at Connellsville is 
insignificant. 

The Youghiogheny Reservoir, downstream of the Deep Creek dam, was completed in October 
1943. It serves multiple purposes and is operated by the U.S. Army Corps of Engineers. The reservoir has 
a capacity of 254,000 acre-feet, which equals 42% of the average annual runoff at the dam, and controls 
a drainage area of 434 square miles, which constitute 33% of the total basin. The length of the river 
between the dam and Connellsville is 29.4 miles. All these facts together suggest that the reservoir can only 
partially control floods at Connellsville. 

Two cases analyzed: present and hypothetical systems 

Two cases of the flood forecast system were analyzed and, accordingly, two sets of parameter 
estimates had to be constructed. The first case describes the present system which is composed of the 
Youghiogheny Dam and the National Weather Service (NWS) river forecasting technology. A flood forecast 
for Connellsville is prepared by routing the project regulated outflow from the dam and superimposing on 
it the predicted runoff from the drainage area between the dam and the forecast point. The second case 
describes a hypothetical system composed of the NWS river forecasting technology but without any 
influence of the Youghiogheny Dam on flood flows. Thus runoff from the entire basin must be predicted, 
as flow at Connellsville is unregulated. 


71 





Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


Data Records and Parameter Estimation 

Historical flood and forecast data were retrieved from the archives of the National Weather Service 
Forecast Office in Pittsburgh. For the present system, with the Youghiogheny Dam, the prior distribution 
of the flood crest was estimated from a record spanning 1943-1986. In these 44 years, 22 flood crests 
exceeded the official flood stage of 12 feet (thus, a flood occurred every two years, on the average). The 
highest flood on record occurred in October 1954 and reached 22 feet. The likelihood functions were 
estimated from a historical joint record of 6 forecasted and actual crests in the period 1984-1986. 

For the hypothetical system, without the Youghiogheny Dam, the prior distribution of flood crests 
was estimated from a record spanning 1910-1942. During these 43 years, 22 flood crests were observed 
above the flood stage of 12 feet. The highest flood during that period occurred in March 1936 and exceeded 
20 feet. Estimation of the likelihood functions presented a challenge since there is no historical joint record 
of forecasted and actual flood crests-a record that would correspond to the modem forecasting technology 
yet without any influence of the Youghiogheny Dam. The theory of sufficient comparisons of forecasts 
systems [Krzysztofowicz 1992] came to the rescue here. It seemed reasonable to assume that systems 
utilizing the same forecasting technology and operated for rivers with similar geomorphologic, hydrologic, 
and climatic characteristics should exhibit similar statistical characteristics of performance. In particular, 
their standardized sufficiency characteristics (SSC) should be similar. [For a definition and properties of 
the SSC see Krzysztofowicz 1992.] A flood forecast system for Milton, Pennsylvania, where river flows 
are unregulated, was taken as an analog. Its SSC was estimated from a historical joint record of forecasted 
and actual flood crests; this record contained 8 forecasts of floods that occurred in the period 1959-1975. 
Next, the variance estimate in the likelihood functions for Connellsville, the case with the dam, was adjusted 
so as to give the SSC for Connellsville the same magnitude as the SSC for Milton. In a sense, we have done 
a "statistical transfer" of a forecast system from Milton to Connellsville. 

Distribution of Damages 

The stage-damage function for Connellsville was estimated according to the methodology of 
Krzysztofowicz and Davis [1983a,b]. A crude inventory of establishments located at various elevations of 
the floodplain was extracted from the River Stage Data form prepared by the National Weather Service and 
the U.S. Geological Survey in May 1990. About 212 establishments were counted in the floodplain and the 
maximum possible damage for the community was estimated to be $10,400,000 at the 1991 price level. 

In order to construct the stage-damage function, the floodplain was discretized into five steps, 
whose elevations range from the flood stage at 12 feet to 20 feet, and the establishments were grouped intro 
three structural categories: two-story house, commercial garage, and commercial store. 

Stage-Damage Function 

For each structural category, there is a unit damage function specifying the fraction of maximum 
possible damage to an establishment which occurs when the depth of flooding, measured from the first-floor 
level, is given. The general form of the unit damage function is polynomial, with the polynomial 
coefficients as given in the technical section for each of the three structural categories. Using all of the 
above information, the stage-damage function for the community may be constructed. 


72 





Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Results of Case Study 

In the case study of Connellsville, Pennsylvania, four different situations with structural and 
nonstructural flood prevention measures are investigated. The expected flood losses in the following four 
cases are calculated: 

(1) expected flood loss in the case with neither a dam nor a flood warning system, 

(2) expected flood loss in the case with a dam and without a flood warning system, 

(3) expected flood loss in the case without a dam and with a flood warning system, and 

(4) expected flood loss in the case with both a dam and a flood warning system. 

When there is a dam, from the historical data the flood crest is fitted by a prior normal distribution, 
and the conditional probability density function of the forecasted flood crest, given the actual crest, is fitted 
by a normal-linear distribution. It can be shown that (a) the marginal probability density function of the 
forecast is of a normal distribution and (b) the posterior distribution density function of the actual crest, 
given the forecasted crest, is of the normal-linear form. 

When there is no dam, from the historical data the flood crest is fitted by a prior normal distribution 
and the conditional probability density function of the forecasted flood crest, given the actual crest, is fitted 
by a normal distribution. It can be shown that (a) the marginal probability density function of the forecast 
is of a normal distribution and (b) the posterior distribution density function of the actual crest, given the 
forecasted crest, is of the normal linear form. 

The four probabilistic measures of the forecasting system can be calculated for given warning 
threshold and zone elevation (Table 4-3). Different values of warning threshold are associated with different 
values of the probabilistic measures: probabilities of correct warning, correct quiet, false warning, and 
missed warning. They thus yield different impacts on the response fraction at the subsequent flood events. 

The unit damage function is fitted from historical data, and the other loss-function parameters are 
set by means of reasonable assumptions described in the technical volume. 

The technical volume gives the calculated values of the expected flood loss without a warning 
system for zone elevation equal to 12 and 14 feet for both cases with a dam and without a dam. It also gives 
the calculated values of the expected flood-loss reduction with full response for various preselected warning 
thresholds for elevations of 12 and 14 feet, in both cases with a dam and without a dam. 


73 





Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


Table 4-3. Probabilistic Measures of the Warning System (Connellsville, Pennsylvania) 



y 

- 12 (with a dam) 


* 

* 

* 

* 


s 

P n (s ,y) 

p io (s -y> 

P 01^ S 

p oo (s ' y) 

11.0 

0.7763117 

0.1021628 

0.0014434 

0.1200821 

11.2 

0.7748482 

0.0879903 

0.0029068 

0.1342546 

11.4 

0.7722843 

0.0735971 

0.0054707 

0.1486478 

11.6 

0.7681055 

0.0594870 

0.0096495 

0.1627579 

11.8 

0.7617464 

0.0462294 

0.0160087 

0.1760156 

12.0 

0.7526836 

0.0343665 

0.0250714 

0.1878784 

12.2 

0.7405316 

0.0243194 

0.0372235 

0.1979255 

12.4 

0.7251226 

0.0163076 

0.0526324 

0.2059374 

12.6 

0.7065377 

0.0103184 

0.0712173 

0.2119266 

12.8 

0.6850762 

0.0061376 

0.0926788 

0.2161073 


y ■ 

■ 14 (with a dam) 


* 

* 

* 

* 

* 

s 

p u (s .y) 

P 10 <s ' y> 

p oi (s ■ y) 

P 00 (s ' y) 

13.6 

0.4459119 

0.1342705 

0.0045577 

0.4152599 

13.8 

0.4421889 

0.1087867 

0.0082806 

0.4407437 

14.0 

0.4363676 

0.0851224 

0.0141020 

0.4644080 

14.2 

0.4278818 

0.0640043 

0.0225878 

0.4855261 

14.4 

0.4163113 

0.0460156 

0.0341583 

0.5035149 

14.6 

0.4014937 

0.0314805 

0.0489759 

0.5180500 

14.8 

0.3835860 

0.0204010 

0.0668836 

0.5291294 

15.0 

0.3630455 

0.0124725 

0.0874241 

0.5370579 

15.2 

0.3405446 

0.0071667 

0.1099250 

0.5423638 

15.4 

0.3168429 

0.0038577 

0.1336266 

0.5456727 


74 




































Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Table 4-3. (continued) 
y - 12 (without a dam) 


* 

* 

★ 

* 

* 

s 

P n ( s -y) 

P 10 (s ' y> 

P 01 (s ' y) 

p oo (s ' y> 

9.6 

0.8598625 

0.0954413 

0.0079126 

0.0367837 

9.8 

0.8577797 

0.0913708 

0.0099954 

0.0408541 

10.0 

0.8552552 

0.0870857 

0.0125198 

0.0451392 

10.2 

0.8522224 

0.0826117 

0.0155527 

0.0496133 

10.4 

0.8486115 

0.0779786 

0.0191636 

0.0542464 

10.6 

0.8443505 

0.0732205 

0.0234246 

0.0590045 

10.8 

0.8393664 

0.0683756 

0.0284086 

0.0638494 

11.0 

0.8335869 

0.0634843 

0.0341881 

0.0687406 

11.2 

0.8269404 

0.0585908 

0.0408347 

0.0736342 

11.4 

0.8193607 

0.0537378 

0.0484143 

0.0784871 


y - 14 (without a dam) 


* 

* 

* 

* 

* 

s 

P n (s ,y) 

P 10 (S •*> 

P 01 (s ' y) 

</i 

v_/ 

o 

o 

Pu 

13.0 

0.5588912 

0.1818558 

0.0335281 

0.2257249 

13.2 

0.5522861 

0.1680066 

0.0401332 

0.2395740 

13.4 

0.5447636 

0.1543220 

0.0476557 

0.2532586 

13.6 

0.5362751 

0.1409070 

0.0561443 

0.2666736 

13.8 

0.5267833 

0.1278629 

0.0656361 

0.2797178 

14.0 

0.5162637 

0.1152843 

0.0761557 

0.2922964 

14.2 

0.5047076 

0.1032567 

0.0877118 

0.3043239 

14.4 

0.4921228 

0.0918543 

0.1002966 

0.3157264 

14.6 

0.4785339 

0.0811387 

0.1138854 

0.3264419 

14.8 

0.4639837 

0.0711576 

0.1284356 

0.3364230 


75 





































Risk-Based Evaluation of 

GLOSSARY OF SYMBOLS 

Part 1. Integration of Flood Warning and Structural Measures 


C E 

D 

E 

F 

f(L) 

U 


U 

h 


L 

Lrd 

L w 

^wo 

M 

MC 

MD 

MR(h - y) 


N 

P(L) 

W 


cost function of evacuation 

flood discharge; used in the frequency-discharge-elevation curves 
flood elevation; used in the discharge-elevation curve 
flood frequency (exceedance probability) 
probability density function of damage L 

conditional expected value of flood damage given exceedance of the flood with 
nonexceedance probability a; measure of the risk of extreme events in the PMRM 

expected value of flood damage 

flood stage 

flood damage (millions $) 

flood loss reduction defmed as the difference between and L, 
flood loss function with a warning system 

flood loss function without a warning system 

number of feasible options involving only flood warning systems for flood mitigation 

maximum evacuation cost to community assuming full response 

maximum possible damage of the community due to flood of the highest magnitude 

unit reduction function specifying the reduction of the maximum flood loss MD when the 
depth of flooding is (h - y) and full response of the community is made (q = 1) 

number of feasible options involving only structural measures for flood mitigation 

probability of flood 

denotes plans incorporating flood warning systems 

77 





Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


y elevation of the floodplain zone under consideration 

a nonexceedance probability that partitions the range of extreme events; used in the definition 

of the conditional expected value f 4 

6(h - y) unit damage function specifying the fraction of MD that occurs when the depth of flooding 

is (h - y) 

0 fraction of the community that responds to a flood warning; response fraction 


Part 2. Multiobjective Decision-Tree Analysis 


a, 

C 

C1,C2,C3 

d™ 

DN1.DN2 

EM 

E’D 

EV1.EV2 

EVE 

$ 

f, 

f 2 ,f 3 

U 

f/ 

fs 

k 


action, or alternative, or option, at a decision node n 

maximum possible loss of property (discrete case); possible loss of lives given no flood 
warning -- linear function of discharge W (continuous case) 

chance nodes in the decision tree 

number of elements in the the set Tf 

do-nothing option in the first and second decision periods, respectively 
expected value 

the Jth averaging-out strategy; for example E 4 denotes the conditional expected value of 
extreme events f„ 

evacuation order in the first and second decision periods, respectively 

expected value of experimentation; difference between expected loss without 
experimentation and expected loss with experimentation 

standard normal distribution function 

cost objective function; balanced with the risk functions f 2 thru f 5 in the PMRM 
conditional expected values 

conditional expected value of the (damage) risk of extreme events 
optimal value of f 4 , see Equation (2.17) 
overall expected value of damage 
dimension of the objective function vector 


78 







Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


L maximum possible loss of lives (discrete case); possible loss of lives given no flood 

warning -- linear function of discharge W (continuous case) 

LN lognormal distribution 

P x cumulative distribution function of X 

p x probability density function of X 

r the vector of objective functions in the decision tree [r,,...,rj 

Tj" set of Pareto optimum alternatives associated with each branch emerging from chance 

node m 

W actual flood level (cfs) 

WA1.WA2 issuing a flood watch in the first and second decision periods, respectively 

X random variable of damage or loss 

a partitioning nonexceedance for the conditional expected value f 4 

a, values of nonexceedance probability that partition the ranges of risk in the PMRM 

pjj values of damage that partition the severity of risk in the PMRM for the y'th policy 

tradeoffs between the cost objective function and the tth risk function 

H mean of the discharge W 

0„ state of nature at node n of the decision tree (also used in unrelated context as 

parameters in the PMRM defined by Equation 2.4) 

o standard deviation of the discharge W 

Part 3. Performance Characteristics of a Flood Warning System 


a,b parameters of the normal-linear likelihood model f 

D detection (Equation 3.7) 

F false warning (Equation 3.7) 

f(s | h,0=l,T=l) probability density of s conditional on the actual crest h, 0 = 1, T = 1 


79 





Risk-Based Evaluation of 

Flood Warning and Preparedness Systems 


FSC 

g(h | 0 = 1) 

ga | e = i) 

h 

LT 

M 

N 

N 

n 

ND 

NF 

PTC 

q 

q(s) 

q* 

ROC 

s 

T 

W 

W*(s) 

y 

y 


forecast sufficiency characteristic, a measure sufficient for comparing any two 
forecasters who produce forecasts of the same variate 

prior probability density function of flood crest given flood occurs 

probability density of A conditional on 0 = 1 

height of actual flood crest 

expected lead time 

missed flood (Equation 3.7) 

expected number of floods per year 

normal probability distribution 

expected number of zone floods per year 

expected number of detections per year for a zone 

expected number of false warnings per year for a zone 

performance tradeoff characteristic, a plot of ND versus NF 

quiet (Equation 3.7) 

P(0 = 1 | s, T = 1), posterior probability of a flood in a given zone 

optimal threshold associated with warning rule W" 
relative operating characteristic, a plot of P(D) versus P(F) 

forecasted flood crest 

trigger indicator: trigger is not observed (T = 0), trigger is observed (T = 1) 

warning rule, w = W(s), where w = 0 and w = 1 denote "do not issue warning" 
and "do issue warning," respectively 

optimal warning rule (of the threshold type) minimizes expected disutility of 
outcomes 

zone elevation 

P(0 = 1 | T = 1), diagnosticity conditional probability 


80 




Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


Ko(s | 0=0,T=l) 

X 


Mh.O,, 


M..O, 

0 

P 

e 


probability density of s conditional on the forecast 0 = 0 and T = 1 

lead time of a warning for a given zone, conditional on hypothesis that zone will be 
flooded 

mean and standard deviation of the prior density g(h | 0 = 1) 
mean and standard deviation of the likelihood function k<, 

zone flood indicator: zone flood does not occur (0 = 0), zone flood occurs (0=1) 

P(T = 1 | 0 = 1), reliability conditional probability 

flood indicator: flood does not occur (0 = 0), flood occurs (0 = 1) 


Part 4. Selection of Optimal Food Warning Threshold 


A.B.C 


C,,Cj 


D(h) 

Doo 

D 01 

Do, 

Dio 

D| 0 

d 10 

D n 

D„ 

d„ 

erf 

f(h | s) 


parameters used in the normal-linear likelihood model (Equations 4.7-4.9) 
constants governing the evolution of the response fraction a 
stage-damage function for a community 

expected loss when no warning is given and no flood occurs (zero) 
expected community property loss without a warning 
expected property loss without a warning conditioned on forecast s 
cost of evacuation in the community 

expected cost of community evacuation conditioned on forecast s 

cost function of evacuation, linear function of response fraction a 

expected community property loss with a warning 

expected property loss with a warning conditioned on forecast s 

loss function with a warning 

standard error function 

posterior distribution of h given a forecast s 


81 







Risk-Based Evaluation of 

Flood 'Wanting and Preparedness Systems 


f(s | h) 

f. 

f. T 

f, T 

u 

g(h) 

h 

k(s) 

MC 

MD 

MR(h - y) 
N 

N (m,o) 

Poo 

P 0I 

P,0 

Pm 

q(s,y) 
r = 1,2,3 
s 
s* 

4> 

y 

a T 

82 


conditional density of s given h 

sura of the expected property loss reductions over the planning horizon 
expected property loss reduction (difference made by warning system) 
expected property loss reduction at stage T 

objective function representing credibility of forecast system: E{a N+1 } 
prior probability density of flood crest h 
flood crest 

marginal probability density of forcast s 

maximum evacuation cost with a full response 

maximum possible damage due to highest flooding with no response 

unit reduction function-reduction of MD when the depth of flooding is (h - y) and a = 1 

number of successive flood events on planning horizon 

normal distribution with mean /x and variance s J 
probability of a correct quiet 

probability of a missed forecast (Type I error) 

probability of a false warning (Type II error) 

probability of a correct warning 

probability that zone of elevation y will be flooded conditioned on forecast s 
structural categories in the floodplain 
forecasted flood crest 

flood warning threshold; warning issued when s * s* 

noninferior decision sequence consisting of the set of warning thresholds for all decision 
periods in the planning horizon 

elevation of a zone in the floodplain 

response fraction of the community in period T 





6(h - y) 

6 r (z) 

<t> 


Risk-Based Evaluation of 

unit damage function specifies the fraction of MD when flood depth is (h - y) 

fraction of maximum possible damage to an establishment that occurs when the depth of 
flooding measured from the first floor level is z 

standard normal distribution function 

mean and standard deviation of the distribution g(h) 


a 

83 




Risk-Based Evaluation of 
Flood Warning and Preparedness Systems 


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