seismic risk estimation

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QUANTIFYING THE UNCERTAINTY IN SEISMIC RISK AND LOSS

ESTIMATION

Patricia Grossi

A DISSERTATION

in

Systems Engineering

Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of

the Requirements for the Degree of Doctor of Philosophy

2000

Supervisor of Dissertation

Graduate Group Chairperson


UMI Number 9976429

Copyright 2000 by Grossi, Patricia

All rights reserved.

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UMIUMI Microform9976429

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P.O. Box 1346 Ann Arbor, Ml 48106-1346


COPYRIGHT

Patricia Grossi

2000


DEDICATION

To my parents

in


ACKNOWLEDGMENTS

The support o f the National Science Foundation through grant CMS97-14401 and the

Lillian Beck Fund in the School of Engineering and Applied Science at the University of

Pennsylvania are both gratefully acknowledged. Additionally, this work has been partially

supported through the NEHRP Graduate Fellowship in Earthquake Hazard Reduction awarded by

the Earthquake Engineering Research Institute under a cooperative program funded by the

Federal Emergency Management Agency.

The support o f numerous individuals is also gratefully acknowledged. This work could

not have been completed without the members o f my dissertation committee, others in industry

working with me as part of the National Science Foundation project, and the respondents to my

survey. First, I would like to thank the members o f my dissertation committee from the

University o f Pennsylvania: Paul Kleindorfer and Howard Kunreuther from the Department of

Operations and Information Management at the Wharton School and G. Anandal ingam and John

Lepore from the Department o f Systems Engineering. I would especially like to thank Robert

Whitman from the Massachusetts Institute o f Technology, who so graciously took the time to

advise me on various aspects of the HAZUS methodology. Also, Weimin Dong of Risk

Management Solutions, Inc. and Scott Lawson of Durham Technologies, Inc., as part o f the

National Science Foundation project, provided the necessary material and support for the creation

of the Scenario Builder and EP Maker software modules used in this work. Finally, the

anonymous structural engineers and contractors in California who took the time to respond to my

expert opinion survey provided me with crucial data for this work.

I would also like to acknowledge the support o f those in the Wharton School and in

industry with whom I have worked on the Managing Catastrophic Risks project, especially those

on the Technical Advisory Committee. From the Wharton School, I would like to thank Steveiv


Levy and Jaideep Hebbar for their support. From industry, I would like to acknowledge Karen

Clark and Nozar Kishi o f Applied Insurance Research, Inc., Dennis Kuzak and Tom Larsen o f

EQE, and Hemant Shah and Don Windeler of Risk Management Solutions, Inc. for their

dedication to the project.

O f course, I must thank my friends, family, and colleagues who have supported me

throughout the years. The help o f Shelley Brown and Denice Gorte in the Department o f Systems

Engineering and the support of my extended relatives in the Grossi and Quinn families, has not

gone unnoticed. Most importantly, the members of my immediate family, Dad, Mom, Jim,

Teresa, Matt, Julia, and Briana, as well as Mohan and Lara, are my inspiration.

v


ABSTRACT

QUANTIFYING THE UNCERTAINTY IN SEISMIC RISK AND LOSS ESTIMATION

Patricia Grossi

Howard Kunreuther

In this work, the role of uncertainty in a probabilistic earthquake loss estimation is

studied. In order to quantify the uncertainty associated with earthquake loss estimation (ELE), a

sensitivity' analysis is completed, assuming alternative estimates for different parameters and

models in the ELE process. The parametric and modeling estimates are varied one-by-one and the

effects on the calculations of average annual loss (AAL) and worst case loss (WCL) are analyzed.

These losses are generated via a loss exceedance probability (EP) curve. This study is unique in

that it uses a regional loss estimation model, HAZUS, with pre-processing and post-processing

software modules to estimate direct economic losses to homeowners and the insurance industry in

the Oakland, California region. A probabilistic seismic hazard analysis (PSHA) is mimicked

through the use o f the pre-processor (Scenario Builder) and post-processor (EP Maker) in the

study. Within this sensitivity analysis, annual recurrence of earthquakes, attenuation models for

ground motion, soil mapping schemes, and exposure and vulnerability parameters for residential

structures are considered. Additionally, techniques to incorporate expert opinion on the

vulnerability of structures, the benefits of structural mitigation, and the costs o f mitigation into

the probabilistic seismic risk assessment are introduced. Conclusions are three-fold. First, the

earthquake loss estimation process is very uncertain, producing estimates of direct economic loss

that are most sensitive to the ground motion attenuation in the Oakland, California region.

Second, the residential structural mitigation studied, bolting a low-rise wood frame structure to its

foundation and bracing its cripple wall, is extremely worthwhile for a homeowner to complete,

using a straightforward cost-benefit approach. Finally, the homeowner will have to cover thevi


majority o f the loss on an average annual basis under various insurance deductible and limit level

schemes.

vii


TABLE OF CONTENTS

COPYRIGHT.............................................................. ii

DEDICATION.................................................................................................................................. iii

ACKNOWLEDGMENTS.............................................................................................................. iv

ABSTRACT____________________________________ vi

LIST OF TABLES.......................................................................................................................... xii

LIST OF ILLUSTRATIONS____________________________________________________xv

1 INTRODUCTION........................................................................................................................ 1

1.1 Overview................................................................................................................................... I

1.2 Scope and Objective.................................................................................................................. 4

1.3 Organization o f Dissertation....................................................................................................13

2 UNCERTAINTY IN EARTHQUAKE LOSS ESTIMATION........................................... 15

2.1 Aleatory versus Epistemic Uncertainty.................................................................................. 15

2.2 Seismic Hazard M odel.......................................................................................................... 17

2.2.1 Previous Studies on PSHA Uncertainty........................................................................ 18

2.2.2 Probabilistic Seismic Hazard Analysis.........................................................................22

2.2.2.1 Seismic Source Determination.................................................................................. 24

2.2.2.2 Recurrence Relationship........................................................................................... 27

2.2.2.3 Estimation o f Ground Motion................................................................................... 30

2.2.2.4 Probability o f Exceedance.........................................................................................33

2.3 Previous Studies on Earthquake Loss Estimation................................................................. 35

2.4 Inventory Exposure Characteristics........................................................................................ 38

2.5 Damage Estimation.................................................................................................................40

2.6 Loss Estimation........................................................................................................................43viii


2.7 Summary................................................................................................................................. 45

3 HAZUS METHODOLOGY AND SOFTWARE..................................................................46

3.1 Scenario Builder......................................................................................................................49

3.1.1 Seismic Source Determination......................................................................................50

3 .1.2 Recurrence Relationship................................................................................................ 53

3.1.3 Estimation o f Ground Motion........................................................................................54

3.1.4 Scenario Builder Input File............................................................................................57

3.2 HAZUS.................................................................................................................................... 59

3.2.1 Ground Motion Methodology........................................................................................59

3.2.1.1 Standardized Response Spectrum............................................................................60

3.2.1.2 Soil Amplification.....................................................................................................63

3.2.2 Inventory Exposure Characteristics.............................................................................. 66

3.2.2.1 Building Capacity Curve.......................................................................................... 68

3.2.3 Direct Physical Damage................................................................................................70

3.2.3.1 Building Fragility Curve.......................................................................................... 72

3.2.4 Direct Economic Losses................................................................................................76

3.3 EP Maker.................................................................................................................................83

4 MODEL CITY AND MITIGATION..................................................................................... 88

4.1 Demographics and Seismic Sources......................................................................................89

4.2 Residential Building Stock and Mitigation Technique........................................................92

4.3 Expert Opinion Incorporation................................................................................................97

4.3.1 Related Research on Vulnerability and Mitigation...................................................... 98

4.3.2 Expert Opinion Surveys............................................................................................... 101

4.3.3 Response Rate and Comments..................................................................................... 104ix


4.3.4 Mean Damage and Cost................................................................................................107

4.3.5 Distribution Fit.............................................................................................................. 115

4.3.6 Structural Fragility Curve Development..................................................................... 121

4.3.6.1 Damage State Probability Matrices........................................................................ 123

4.3.6.2 Cumulative Lognormal Distributions................................................................... 125

4.3.6.3 Conversion to Peak Ground Acceleration.............................................................. 129

4.3.6.4 Intersection o f Demand Spectrum and Capacity Curve........................................130

4.3.7 Discussion.................................................................................................................... 139

5 PARAMETERS IN SENSITIVITY ANALYSIS............................................................... 145

5.1 Seismic Hazard Parameters..................................................................................................147

5.1.1 Earthquake Recurrence................................................................................................ 149

5.1.2 Ground Motion Attenuation....................................................................................... 155

5.1.3 Soil Mapping Schemes............................................................................................... 156

5.2 Inventory, Damage, and Loss Parameters............................................................................160

5.2.1 Inventory Exposure...................................................................................................... 161

5.2.2 Fragility Curves............................................................................................................162

6 RESULTS.................................................................................................................................168

6.1 Range of Uncertainty............................................................................................................ 168

6.2 Expert Opinion Incorporation for Mitigation....................................................................182

6.3 Cost-Effectiveness o f Mitigation......................................................................................... 187

6.4 Earthquake Insurance........................................................................................................... 192

7 CONCLUSIONS..................................................................................................................... 202

7.1 Summary...............................................................................................................................202

7.2 Future W ork......................................................................................................................... 206x


8 Appendices................................................................................................................................. 207

8.1 Earthquake Measures............................................................................................................ 207

8.2 Attenuation Relationships.....................................................................................................210

8.3 EP Maker C ode......................................................................................................................213

8.4 Scenario Builder Input File................................................................................................. 219

8.5 Benefits Questionnaire.........................................................................................................221

8.6 Costs Questionnaire...............................................................................................................224

8.7 Benefits Data..........................................................................................................................227

8.8 Costs Data..............................................................................................................................231

8.9 Distributions...........................................................................................................................232

8.10 Recurrence Data...................................................................................................................234

8.11 Soils Data............................................................................................................................. 235

8.12 Exposure Data...................................................................................................................... 238

8.13 Summary o f Average Annual Loss.................................................................................... 241

9 Bibliography............................................................................................................................... 242

xi


LIST OF TABLES

Table 1.1 Homeowner Losses as a Function of Mitigation and Insurance............................. 12

Table 1.2 Insurer Losses as a Function of Structural Mitigation. ____........___........___ 13

Table 3.1 Event Loss Table.............................................................................................................. 47

Table 3.2 Fault Segment and Property Information.................................................................... 51

Table 3 3 Scenario Builder Input File............___......................................... 58

Table 3.4 Regression Coefficients for Fault Rupture................................................................... 59

Table 3.5 NEHRP Site Classes.........................................................................................................64

Table 3.6 NEHRP Soil Amplification Factors...............................................................................65

Table 3.7 Occupancy Classes and Model Building Types............................................................67

Table 3.8 Drift Ratios and Spectral Displacement for Structural Damage.............................. 73

Table 3.9 Structural Fragility Curves.............................................................................................75

Table 3.10 Structure Occupancy Mapping.................................................................................... 76

Table 3.11 Types of Direct Economic Loss to Residential Structures.......................................77

Table 3.12 Time Independent Direct Economic Loss Parameters............................................. 80

Table 3.13 Time Dependent Direct Economic Loss Parameters................................................ 82

Table 3.14 Aggregation of Direct Economic Losses..................................................................... 84

Table 4.1 Oakland, California Region Demographics.................................................................90

Table 4.2 Default RES1 Occupancy Scheme.................................................................................. 93

Table 4.3 Updated RES1 Occupancy Scheme................................................................................96

Table 4.4(a) RES1 Occupancy Scheme for Analysis (Before Mitigation)................................ 96

Table 4.4(b) RES1 Occupancy Scheme for Analysis (After Mitigation).................................. 96

Table 4.5 Mean Damage Factors.................................................................................................... 100

Table 4.6 Costs o f Mitigation.......................................................................................................... 101xii


Table 4.7 Survey Respondents' Statistics.....................................................................................104

Table 4.8 Cases for Mathematical Aggregation.......................................................................... 108

Table 4.9 Benefits Survey Statistics of MDF.............................................................................. I l l

Table 4.10 Costs Survey Statistics................................................................................................. 114

Table 4.11 Beta Distributions Parameters for Damage Data................................................... 117

Table 4.12 Best Distribution Fit Parameters for Damage Data...............................................119

Table 4.13 Best Distribution Fit Parameters for Cost Data..................................................... 120

Table 4.14 Damage Factor State Limits.......................................................................................124

Table 4.15 Damage State Probability Matrices.......................................................................... 125

Table 4.16 Cumulative Damage State Probability Matrices.................................................... 126

Table 4.17 Statistics of Cumulative Lognormal Distributions (MMI)....................................128

Table 4.18 MMI to PGA Conversion Table................................................................................ 130

Table 4.19 Statistics of Cumulative Lognormal Distributions (PGA).................................... 130

Table 4.20 Response Spectrum Demand Parameter Values..................................................... 134

Table 4.21 Capacity Curve Parameter Values............................................................................ 135

Table 4.22 Capacity Curve Parameter Values............................................................................ 136

Table 4.23 Structural Fragility Curve Parameters.................................................................... 138

Table 5.1 Seismic Fault Source Parameters................................................................................ 150

Table 5.2 Characteristic Recurrence............................................................................................ 151

Table 53 Exponential Recurrence................................................................................................ 152

Table 5.4 Structural Fragility Curve Parameters...................................................................... 164

Table 5.5 Nonstructural Fragility Curve Parameters................................................................ 166

Table 6.1 Cases for Range of Uncertainty Calculations.............................................................170

Table 6.2 Notation for Range of Uncertainty Calculations...................................................... 171xiii


Table 6.3 Summary of Loss Estimates for Case 2 .......... 175

Table 6.4 Summary of Losses for Case 3........................................... 178

Table 6.5 Breakdown of Average Annual Loss........................................................................ 180

Table 6.6 Best Distribution Fit Parameters for Cost Data........................................................ 183

Table 6.7 Effects o f Mitigation....................................................................................................184

Table 6.8 Cost-Benefit Analysis of Mitigation...... .......___...___ .... ----------- 189

Table 6.9 Simulation Results...........................................................................................................191

Table 6.10 Homeowner Loss as a Function o f Mitigation and Insurance...............................192

Table 6.11 Insurer Loss as a Function of Structural Mitigation............................................... 193

Table 6.12 Parameters for Analysis............................................................................................... 194

Table 6.13 Homeowner Average Annual Loss without Insurance............................................195

Table 6.14 Homeowner Average Annual Loss with Insurance and No Mitigation......... 197

Table 6.15 Homeowner Average Annual Loss with Insurance and Mitigation.....................198

Table 6.16 Insurer Losses................................................................................................................200

Table 6.17 Percentage Reduction in Insurer Losses................................................................... 201

Table 7.1 Ranking of Parameters and Models by Range of Uncertainty............................... 203

xiv


1 INTRODUCTION

1.1 Overview

In 1989, two events occurred that had a drastic impact on both the insurance industry and

the political climate in the United States. Late on September 21, 1989, Hurricane Hugo hit the

coast of South Carolina, devastating the towns o f Charleston and Myrtle Beach. Insured loss

estimates totaled $4.2 billion before the storm moved through North Carolina the next day

(Insurance Information Institute 2000). Less than a month later, on October 17, 1989, the Loma

Prieta Earthquake hit with a Richter Magnitude o f 7.1 near the town o f Santa Cruz, California.

Property damage was estimated at $6 billion to the surrounding Bay Area (Stover and Coffman

1993). In order to remain solvent, the insurance industry realized that it needed a better way to

estimate and manage the losses associated with such natural disasters. Moreover, the Federal

Emergency Management Agency (FEMA) recognized a need for better catastrophic loss

estimates for mitigation and emergency planning purposes.

As a result, over the course of the past decade, advanced tools have emerged that allow

insurance companies and agencies of the government to more accurately assess their catastrophic

risk exposures. These software tools utilize advances in Information Technology (IT) and

Geographic Information Systems (GIS). With the ability to store and manage vast amounts of

spatially referenced information, GIS is an ideal environment for conducting catastrophic hazard

and loss studies for large regions. While it is true that catastrophe loss studies were done for

twenty years prior to this time (eg. earthquake loss estimates in Steinbrugge 1982), this

advancement in computer technology has created an easier, more cost-effective way to perform

these studies.

1


These software tools, known as catastrophic risk models, were developed in two separate

arenas (Figure 1.1). First, private companies developed models for insurance companies to

estimate their portfolio losses and individuals to estimate their site-specific losses from either a

probabilistic seismic hazard analysis (PSHA) or a deterministic earthquake scenario. Second, the

federal government developed a regional loss estimation model (HAZUS) to estimate monetary

losses as well as other types o f losses (eg. casualties and shelter requirements) from an earthquake

event.

GeographicInformation

Systems

HAZUS

InformationTechnology

AIREQECAT

RMS

Private Companies

Catastrophic Risk Modeling

US Government

Figure 1.1 Catastrophic Risk Modeling Development

A few private companies have emerged as leaders in the field of catastrophic risk

modeling. Those familiar to the author through joint work with the Wharton School include

Applied Insurance Research, Inc. (AIR), EQECAT, and Risk Management Solutions, Inc. (RMS).

Each firm has its own software package, analyzing the economic effects from both earthquakes

and hurricanes in the United States for insurance and reinsurance companies. Furthermore, in

1997, FEMA released the first version of its own software estimating potential earthquake losses

in the United States, called HAZUS (Hazards U.S.)- Current development is underway to

estimate wind (eg. hurricane) and flood hazards using HAZUS.

2


While the private industrys software and the HAZUS software are intended for different

audiences, each utilizes the same general methodology to analyze catastrophic earthquake losses.

The earthquake loss estimation (ELE) methodology is comprised o f four basic stages: (1) define

the earthquake hazard, (2) define the inventory characteristics, (3) estimate the inventory damage,

and (4) calculate the economic losses (Figure 1.2).

Calculate Economic Loss(Expected Loss or WCL to Insurer and Owner)

Define the Earthquake Hazard(Seismic Sources, Recurrence,

Attenuation, Soils)

Define the Inventory Characteristics(Structure Location, Value, Year Built,

Construction Class, etc.)

Estimate Inventory Damage(Through Historical Loss Data,

Engineering Data Expert Opinion)

Figure 1.2 Steps of Earthquake Loss Estimation

There are numerous instances in the earthquake loss estimation procedure where

uncertainty plays a role. Uncertainty is classified by leaders in the field o f seismic hazard analysis

as either aleatory (i.e. randomness) or epistemic (i.e. lack of knowledge) in nature (Budnitz et al.

1997). Furthermore, uncertainty has been categorized into modeling uncertainty, due to

differences between the process being modeled and the simplified model used for analysis, and

parametric uncertainty, due to differences between actual values o f parameters and estimates for

analysis. For the four-stage earthquake loss estimation process, there is uncertainty: in the

seismologicai data used and its interpretation to define the earthquake hazard, in the exposure

data and vulnerability functions used for damage estimation, and in the costs used in determining

3


losses. In general, limited scientific information, lower quality data, or limited engineering

information results in greater variability of expected losses.

Understanding the uncertainty associated with earthquake losses is very important.

Similar to other low-probability, high-consequence (LPHC) events, the problem arises when

decisions need to be made. Decision-making should not be based solely on one average annual

estimate or expected value o f loss, but should consider the variability associated with the

estimate. For example, standard deviations, variances or other statistical measures o f spread

that loss can be from the average are important in decision-making. How widely dispersed the

losses are in a distribution is a key ingredient to making a well-informed, correct decision. If the

decision-maker does not appreciate the complexity of the problem and chooses to ignore the

uncertainties involved, decisions can be made based only on the expected value o f loss and

difficulties can result.

For example, an emergency planner in the city o f Oakland, California can use the

HAZUS software to estimate the number o f displaced households in the city for post-earthquake

event planning. If he bases his decision only the best or mean estimate o f red tagged (i.e. unsafe)

homes, though, a number of residents could remain homeless for longer than necessary after a

severe earthquake. In addition, an insurer can use one of the private industrys software packages

to analyze his insured losses in a significant earthquake event. If he thinks in terms of the annual

expected loss only, he could stand to lose his business through insolvency.

1.2 Scope and Objective

In this work, the role o f uncertainty in a probabilistic earthquake loss estimation

methodology is studied. In order to quantify the uncertainty associated with earthquake loss

estimation (ELE), a sensitivity analysis is completed, assuming alternative estimates for different

parameters and models in the ELE process. The parametric and modeling estimates are varied4


one-by-one and the effects on the calculations o f expected and worst case losses are analyzed.

These losses are generated via a loss exceedance probability curve. A loss exceedance probability

curve, EP(L), is a graphical representation o f the probability that a certain level o f loss will be

exceeded on an annual basis (Equation 1.1 and Figure 1.3).

EP(L) = P(Loss > Z,) = 1 F(L) (1.1)

Probability of Exceedance

EP(L)

AAL

probability = 1 %

Loss (in Dollars) WCL

Figure 1J Loss Exceedance Probability Curve

In Equation 1.1, F(L) denotes the cumulative probability function for the loss or

P(Loss < L) and the loss exceedance probability curve follows simply as 1 - F(L). In Figure 1.3,

it is clear that as the probability o f exceedance increases, expected losses will decrease.

Additionally, two other terms are graphically represented in Figure 1.3: AAL and WCL. AAL is

the expected loss on an annual basis or Average Annual Loss, and it is the area under the loss

exceedance probability curve. WCL is the Worst Case Loss, defined as the loss expected one

percent (1%) o f the time on an annual basis. It is the loss toward the tail end of the curve.1

1 Note that the term worst case loss, rather than probable maximum loss (PML), is used. The term PML is a well-defined concept in the literature (Steinbrugge 1982) and this distinction is made to avoid confusion.

5


The goal o f this work is to ascertain the Range o f Uncertainty of Loss, L, over the

parametric and modeling assumptions, X, denoted RU[L(X)]. L is generalized as a real-value

performance function defined on X. Further, X is a range of information states over which

parametric and modeling values vary. In its generalized form, the Range of Uncertainty of/, over

A'is defined in Equation 1.2.

/?[/[L(Jr)]=[M axZ(x)| x e X ] - [Min L(x) | x e X ] (1.2)

In this work, however, the information states, x e X, are limited to two states, defined as

default and updated. The default state is one in which the HAZUS default value is used in the

analysis. The updated state is one in which a new parameter or model value is incorporated into

the analysis (See Chapter Five). Also, L is in terms o f monetary losses. In its simplest form, in

which A'has two information states, {x

This study is unique in that it will use a regional loss estimation model, HAZUS, with

pre-processing and post-processing software modules to estimate direct economic losses to

homeowners and the insurance industry in the Oakland, California region. A probabilistic seismic

hazard analysis (PSHA) is mimicked through the use o f the pre-processor (Scenario Builder) and

post-processor (EP Maker) in the study. Due to the proprietary nature of the private companies

software packages, the public domain HAZUS software and methodology is utilized.

So, taking into consideration a clearly defined list o f parametric and modeling

uncertainties, in this research, there are four separate questions being addressed.

1. Which parameters and models in the earthquake loss estimation process give rise to the

most uncertainty?

First and most importantly, the primary goal is to discover the sensitivity o f the economic

loss to the parametric and modeling uncertainties in the earthquake loss estimation (ELE) process.

If one can rank these uncertainties by their influence on the loss, the most appropriate areas of

future research and data collection can be recommended to reduce the uncertainty in ELE. In

other words, the sensitivity analysis completed will offer insight into the magnitude of losses to

changes in the values of uncertain models and parameters. Moreover, some insights into how

various parameters and models affect each other will be gained.

The largest absolute difference or RU value will define the parametric or modeling

uncertainty most influential in the earthquake loss estimation process and will be ranked first. The

parametric and modeling uncertainties considered in the analysis include seismic hazard

estimation parameters and models (earthquake recurrence, ground motion attenuation, soil

mapping schemes) and inventory, damage, and loss estimation parameters and models (inventory

exposure of the residential building stock and fragility of certain residential structures before and

after mitigation).7


The Range of Uncertainty, in its barest form, considers losses to society for a static

building stock. This study, however, incorporates earthquake mitigation in the form o f structural

retrofit o f residential buildings and residential earthquake insurance. Therefore, the RU values can

be dependent on the stakeholders, the building fragility state, and the use of insurance. Thus,

Equation 1.3 is restated in Equation 1.4 to reflect these dependencies.

RU[L{X))k = | L{xx (/, j) ) - L(x0 (/, J)) | (1.4)

Specifically stakeholders, k, denote either the homeowner {HO) or Insurer (/); fragility

states, /, denote no structural mitigation {0) or structural mitigation (/); and insurance use,./,

indicates no insurance is in place (0) or insurance is utilized (/). In this way, each RU value

reflects the incorporation o f mitigation and insurance {eg. 22 = 4 combinations for each

stakeholder). This leads to question two.

2. How does one define a param eter in the earthquake loss estimation process so it reflects all

the information provided by experts? How does one define a parameter in a cost-benefit

analysis so it reflects all the information provided by experts?

Expert opinion incorporation is extremely important in the earthquake loss estimation

process and subsequent cost-benefit analyses of the stakeholders in the analysis. In determining

the updated parameters in the fragility {eg. damage) curves utilized in the HAZUS

methodology, the incorporation o f expert opinion into the software is addressed. Specifically, the

two parameters that define a cumulative lognormal structural fragility curve in HAZUS are

estimated using the expert opinion o f structural engineers. These are S D,ds < which is the median

value o f spectral displacement at which building reaches threshold o f structural damage sta ted ,

and PDfc > {he standard deviation of the normal logarithm of spectral displacement o f damage

state ds (See Section 3.2.3.1 for more details).

8


These parameters are defined for pre-1940 low-rise wood frame residential structures

before and after structural mitigation. Structural mitigation is comprised of bolting the iow-rise

wood frame structures to their foundation and bracing their cripple walls. The five-step process to

incorporate expert opinion on the fragility o f these older wood frame homes and the benefits of

this structural mitigation technique (i.e. reduction in damage) into HAZUS is presented.

The change in the annual benefits o f mitigation, AB, due to the incorporation of new

knowledge o f the fragility, x h is the difference between the absolute value o f the annual benefits

of mitigation in this updated information state and those in the default state, x0. In Equation 1.5,

considering benefits to society and no insurance (j = 0), i = 0 reflects no mitigation and / = 1

reflects structural mitigation (i.e. a Boolean indicator of zero or one).

A = \(AAL(xi(0.0))-AAL(xi(l.0))l - \ (AAL(x0(0,0)) - AAL(x0(l.O)) | (1.5)

Furthermore, the expert opinion o f contractors experienced in this particular residential

earthquake retrofit technique (i.e. bracing and bolting) is incorporated into the homeowner cost-

benefit analysis. Specifically, the estimated cost for the contractor to retrofit an older wood frame

home, Crotai, or the homeowner to retrofit the structure himself, CKlalenai, is presented and utilized

in the cost-benefit analysis. These costs, in terms of upfront dollar values, have sample mean

values, with a distribution surrounding the sample mean for the cost-benefit analysis (See Section

4.3.4 and 4.3.5). This leads to question three.

3. Is it beneficial fo r a homeowner in the Oakland, California to mitigate fo r earthquake

hazard?

In performing a sensitivity analysis on the parameters and models used in the earthquake

loss estimation process, the homeowners losses in all cases can be determined. Considering the

case where insurance is not utilized (J = 0), the RU value for the homeowner is different,

9


depending on whether structural mitigation is in place (Equations 1.6(a) and 1.6(b)). In the

equation, / = 0 indicates no mitigation and / = 1 indicates structural mitigation. One of these

scenarios is chosen for calculation of the RU.

RU[L{X)} h o = | L(xx( 0.0)) - L(x0( 0,0)) | (1.6 (a))

RU[L(X)] h o = I L(x\(\.0)) - L(x q ( \ ,0J) | ( 1 .6 (b))

Further, the point at which it is beneficial for the homeowner to mitigate can be

established using a net present value (NPV) calculation (Equation 1.7(a)). In this formulation, the

upfront cost o f mitigation, C, is compared with the annual expected benefits o f mitigation, 4 7 ,

over the lifetime o f the structure, T.

NPV = -C 0 + (1.7(a)) t=\ ( 1 + r)

The upfront cost o f mitigation, CQ, is either Crotai or CUalenal, as previously defined. The

annual expected benefits of mitigation, BAh are simply the average annual loss (AAL) difference

between the unmitigated and mitigated losses using the updated information state, Jt/, or the

default information state, xo (Equation 1.7(b)).

BtM = [AAL(xx( 0,0)) -AAL(xx(\,0))]h o or [AAL(x0(0.0)) - AAL(x0(\,0))JHO (1.7(b))

Finally, r is the discount rate and T is the lifetime of the structure (i.e. the time horizon

ranges from / = 1 to 7). Whenever the NPV in Equation 1.7 is greater than or equal to zero, it will

be beneficial to the homeowner to mitigate.

In the HAZUS methodology, direct benefits are defined as the reduction in direct

economic losses (eg. repair and replacement costs, building content losses, relocation expenses).

While indirect benefits can also be defined (i.e. reduction in induced damage), direct benefits are

the focus o f this study. Framing the analysis in this way, this work assesses the robustness o f a

10


specific residential structural mitigation technique (i.e. bolting and bracing), and the cases in

which mitigation is monetarily beneficial are assessed. Ideally, all cases will be beneficial to the

homeowner for a reasonable life o f the structure.

4. Considering residential earthquake insurance, what are the expected losses to the

homeowner and the residential insurer in the Oakland, California area?

Finally, in performing a sensitivity analysis on the parameters and models used in the loss

estimation process, the homeowners losses and primary insurers losses when insurance is

purchased (j = I) can be determined. The RU value for the primary insurer will be different when

the structures are mitigated (/ = I) and when they are unmitigated (/ = 0), as in the case o f the

homeowner (Equations 1.8(a) and 1.8(b)).

RU[L(X)] i = | L(x$0,l)) - L(xq(Q,$)\ (1.8(a))

RU[L(X)] i = | L(x$\,$) - L(x0(\.\))\ (1.8(b))

Considering various deductible and limit level schemes, the expected losses to

homeowners and the insurance industry in the region are calculated. The reduction in losses due

to mitigation is incorporated in the analysis to ascertain the interaction effects o f the structural

mitigation technique and residential earthquake insurance with the uncertainty in the ELE

process.

In Table 1.1, the annual losses to the homeowner are laid out in terms o f the expected

losses, AAL(x\(iJ))ffQ or AAL(xft(iJ))h q , the insurance premiums paid by the homeowner,

(1 + /y ) - AAL(x\(i,j)) or (1 + /y ) AAL(xq('i, j)~), and the cost of mitigation, a y -C0 . Expected

losses are calculated using the updated information state,x h or default information state, x0. with

mitigation (i.e. i = /) or without mitigation (i.e. i = 0), and with insurance (i.e .j = 1) or without

insurance (i.e .j = 0).

II


Moreover, premiums paid by the homeowner for residential earthquake insurance are

proportional to the Annual Average Loss (AAL) for the property covered and then multiplied by

a loading factor, If, to reflect the administrative costs associated with insurance marketing and

claims settlement. In this case, the loading factor,//, is taken as 0.5. In other words, premiums are

1.5 times the total AAL to the residence. Finally, the costs of mitigation,C0, are multiplied by a

factor, a/, to convert the upfront costs to annualized costs over the lifetime of the structure

T{a f = ^ +^ ). For example, if T= 30 years and r = 10%, a/= 0.106.

(1 + r) -1

Table 1.1 Homeowner Losses as a Function of Mitigation and Insurance

Losses to Homeowner

Insurance(/ = /)

No Insurance

Table 1.2 Insurer Losses as a Function of Structural Mitigation

Losses to Insurer Insurance(/ = /)

No Insurance

detail in this chapter. The updated structural fragility curves developed for use in HAZUS are

presented.

Chapter Five summarizes the parameters used in this sensitivity analysis. Details on the

default and updated information states of earthquake recurrence, attenuation relationships,

soil databases, building stock exposure, and fragility of structures before and after mitigation are

all clearly defined. In all, sixty-four (26 = 64) HAZUS runs are completed for input into the EP

Maker and loss exceedance probability curves are generated under the different states of

information.

Chapter Six presents the results o f this analysis. The impact o f uncertainty, mitigation,

and deductible and limit levels on the homeowner and the residential insurer are presented. The

average annual loss (AAL) and worst case loss (WCL) are emphasized. Moreover, the ranking of

the importance of the parameters and models in this analysis is given.

Finally, Chapter Seven summarizes the conclusions and suggests future developments of

this work.

14


2 UNCERTAINTY IN EARTHQUAKE LOSS ESTIMATION

The earthquake loss estimation process can be delineated in four basic stages: define the

earthquake hazard, define the inventory characteristics, estimate the inventory damage, and

calculate the economic loss. Each step gives rise to much uncertainty. Completed research into

the uncertainty in this area is broken down into two separate sections: estimating the uncertainty

in probabilistic seismic hazard analysis (PSHA) and estimating the uncertainty in earthquake loss

estimation (ELE). A distinction between the two should be emphasized. Probabilistic seismic

hazard analysis is the determination o f the likelihood that a defined level o f ground motion will

be exceeded at a site in a certain time period. Earthquake loss estimation is the determination of

the losses for a region or a site in question for a certain time period using either a deterministic or

probabilistic seismic hazard analysis. Much has been published on the uncertainty in the

parameters and models used in seismic hazard analysis, but little has been published on the

uncertainty in the rest of the earthquake loss estimation process. This chapter reviews the

earthquake loss estimation process in detail and the uncertainty involved in the methodological

process. First, a general discussion of aleatory and epistemic uncertainty is given.

2.1 Aleatory versus Epistemic Uncertainty

In a report published by leaders in the field of probabilistic seismic hazard analysis

(Budnitz et al. 1997), uncertainty is defined as either aleatory or epistemic. Aleatory uncertainty

is "uncertainty that is inherent to the unpredictable nature of future events. It represents unique

details of source, path, and site response that cannot be quantified before the earthquake occurs.

Given a model, one cannot reduce the aleatory uncertainty by collection o f additional

information. One may be able, however, to quantify the aleatory uncertainty better by using

additional data." Epistemic uncertainty is uncertainty that is due to incomplete knowledge and

15


data about the physics o f the earthquake process. In principle, epistemic uncertainty can be

reduced by the collection o f additional information. To extend these definitions to the rest o f the

earthquake loss estimation process: there are unique details in the vulnerability functions used for

damage estimation that cannot be quantified before an earthquake occurs (aleatory uncertainty).

Additionally, there is incomplete knowledge about the inventory site characteristics or exposure

data used in the analysis (epistemic uncertainty).

Besides this distinction between aleatory and epistemic uncertainty, a differentiation

between modeling uncertainty and parametric uncertainty in PSHA is made in Budnitz (1997).

Modeling uncertainty represents differences between the actual physical process that generates

the strong earthquake ground motions and the simplified model used to predict ground motions.

Modeling uncertainty is estimated by comparing model predictions to actual, observed ground

motions. Parametric uncertainty represents uncertainty in the values of model parameters in

future earthquakes. Parametric uncertainty is quantified by observing the variation in parameters

inferred for several earthquakes and/or several recordings. Again, extending these definitions to

the rest of earthquake loss estimation: there are differences between the actual physical damage to

structures and the simplified model used to predict damage (modeling uncertainty), as well as

uncertainty in the values o f parameters estimating repair and replacement cost from future

earthquakes (parametric uncertainty). Both modeling and parametric uncertainties contain

aleatory and epistemic uncertainty.

While the advantage o f differentiating between aleatory and epistemic uncertainty in an

analysis is clear {i.e. only epistemic uncertainty can be reduced), the necessity o f distinguishing

between aleatory and epistemic uncertainty is not. As Thomas Hanks and C. Allin Cornell state

(Hanks and Cornell, 1994):

16


...epistemic and aleatory uncertainties are fixed neither in space (across a range

of models existing in 1997, say) nor in time. What is aleatory uncertainty in one

model can be epistemic uncertainty in another model, at least in part. And what

appears to be aleatory uncertainty at the present time may be cast, at least in part,

into epistemic uncertainty at a later date. As a matter o f practical reality, the trick

is to make sure that uncertainties are neither ignored nor double counted. The

possibilities o f doing so with parametrically complex models are large.

Agreeing with this reasoning, the analysis presented herein is careful not to ignore

uncertainties. The handling of the sources o f uncertainty is clearly outlined in a logic tree

framework. Moreover, the framing of the problem to incorporate updated information states is

implicitly considering epistemic uncertainty, while aleatory uncertainty is ingrained in the

software methodology itself. With this clear, the next step is to determine which parameters and

models used in earthquake loss estimation give rise to uncertainty. This begins with a look at the

seismic hazard model.

2.2 Seism ic Hazard Model

The first step in the earthquake loss estimation process is the determination of the seismic

hazard. The hazard can either be defined as deterministic or probabilistic. Deterministic

earthquake hazards are defined primarily by their magnitude and epicenter location. These

hazards include both historical events and user-defined events {i.e. scenario-based events). An

historical event is a chronicled earthquake occurrence, such as the Northridge earthquake of

January 17, 1994. A user-defined event is a hypothetical event chosen by the user based on an

arbitrary choice of earthquake epicenter along a known fault or in an area source.

Alternatively, a probabilistic hazard is considered. In a probabilistic seismic hazard

analysis (PSHA), all possible seismic sources locations and geometries are determined, the17


maximum magnitude expected from each source is estimated, and the recurrence model or

frequency of earthquake events for each source is obtained. In this study, a series o f carefully

selected scenario-based events on established sources are used to mimic a full-blown PSHA (See

Chapter Three).

Whether the hazard is defined as deterministic or probabilistic, the last portion needed to

define the hazard is the attenuation relationship or ground motion expected at certain distances

from the source for various soil types. In a probabilistic analysis, a further step is to generate a

ground motion exceedance probability curve, designating the probability of exceeding a particular

ground motion level at a certain site.

There are numerous instances in the seismic hazard estimation stage where uncertainty is

prevalent. The estimation o f parameters and choice o f models to utilize are based on the expert

opinions o f geologists and seismologists, the manipulation of historical earthquake catalogue

data, and assumptions in empirically based attenuation and recurrence relationships. Each

assumption plays a role in the overall uncertainty in the process.

2.2.1 Previous Studies on PSHA Uncertainty

In the I980s, two probabilistic seismic hazard analysis (PSHA) studies were performed,

estimating the seismic hazard for nuclear power plant sites in the central and eastern United

States (McGuire et al. 1989; Bemreuter et al. 1989). Both documents produced similar hazard

curves for the relative seismic hazard, but had drastically different estimates o f absolute hazard

levels for several plant sites. Because of the discrepancy, much controversy has surrounded

estimates of seismic hazard. Since this time, the controversy has extended to determining seismic

hazard through PSHA not only for hazardous facilities in the central and eastern United States,

but for other structures across the country. Studies completed on the uncertainty encountered in a

probabilistic seismic hazard analysis have been either qualitative or quantitative in nature.18


Qualitatively, in April o f 1997, a study was published addressing the debate over the

appropriate way in which to conduct a probabilistic seismic hazard analysis (Budnitz et al. 1997).

It was a comprehensive study by a group of seven individuals, known as the Senior Seismic

Hazard Analysis Committee (SSHAC), for the U.S. Nuclear Regulatory Commission, the U.S.

Department of Energy, and the Electric Power Research Institute. Within this document, the issue

of uncertainty is addressed in terms of a PSHA, and a distinction is made between aleatory (i.e.

random) and epistemic (i.e. lack of knowledge) uncertainty, as summarized in Section 2.1.

The SSHAC define the primary objectives o f a well-done PSHA as: proper and full

incorporation of uncertainties, inclusion of a range of diverse technical interpretations, and

consideration o f site-specific knowledge and data sets. Further, complete documentation o f the

process and results, clear responsibility for the conduct of the study, and a proper peer review are

all necessary. They recommend that the inputs into a PSHA be derived either using a Technical

Integrator (TI) approach or a Technical Facilitator/Integrator (TFI) approach. These approaches

assert that one individual, a TI or TFI, is responsible for incorporating and representing the views

of the entire scientific community on the technical issue of interest. They give examples of a

number of techniques that could be utilized in aggregating expert opinions for seismic source

determination or ground motion estimates. In essence, the SSHAC develop a systematic way in

which to incorporate subjective information into a model of seismic hazard. In all aspects o f this

dissertation, every effort is made to follow the guidelines o f the SSHAC report. More

specifically, the author acts as the Technical Integrator in all aspects of the study.

As for quantitative analyses completed on the uncertainty in PSHA, the first notable

study was completed in the early 1980s on the uncertainties associated with the seismic hazard in

the Northern California Bay Area region (McGuire and Shedlock 1981). Five parameters were

varied, using a discrete logic tree. The logic tree included: two interpretations o f the mean rupture19


length, three equations for estimating the mean acceleration (ground motion), three estimates of

the Richter b-value (see Equation 2.4 in Section 2.2.2.2), three estimates of the maximum

magnitude, and three estimates o f the expected events per year. These parameters were varied

over a fault system in the Bay Area that included nineteen faults. The authors conclude that the

coefficient of variation of the 500-year acceleration ranges from 0.2 to 0.4 in the Northern

California region. Additionally, they note that determining the largest sources o f uncertainty in

seismic hazard analysis should be done. In this way, the most appropriate areas for future

research and data collection can be recommended to reduce the uncertainty associated with

seismic hazard analysis. They do not, however, indicate their estimates of the most influential

parameters in the uncertainty analysis.

More recent publications on quantifying the uncertainty in a probabilistic seismic hazard

analysis are a series o f papers in the Bulletin o f the Seismological Society o f America by members

o f the California Department o f Conservation's Division o f Mines and Geology (CDMG). In

these papers, they assess the uncertainty of ground shaking in the Los Angeles, Ventura, and

Orange Counties following the Northridge earthquake (Peterson et al. 1996a; Cao et al. 1996;

Cramer et al. 1996).

In the first of these publications (Peterson et al. 1996a), representatives o f the CDMG

update the earthquake source model o f the Southern California Earthquake Center (SCEC) with

new seismological information and present probabilistic seismic hazard maps (10% probability of

exceedance in 50 years) at two locations, Los Angeles and Northridge. These hazard maps

represent the ground motion at the two sites, incorporating the statistical uncertainty o f a number

o f parameters in the analysis. Specifically, they assumed uncertainty in the fault length, fault

width, slip rate, shear modulus, and recurrence b-value, as well as the uncertainty associated with

the moment-magnitude, magnitude-rupture, attenuation and magnitude distribution relationships.20


In their analysis, they assessed the impact of each of these parameters on the final ground

motion estimate, asserting that the slip rate, moment-magnitude relationship, magnitude

distribution, magnitude-rupture length relationship, and attenuation relationship contribute most

to the overall uncertainty at the Los Angeles and Northridge sites. Furthermore, they state that, at

these two locations, earthquake magnitudes between 5.0 and 8.0 and epicenter distances of less

than 60 km contribute the most to the earthquake hazard. While this final conclusion may be an

obvious one (Le. epicenters closer to the site will be more damaging), this is an important point to

keep in mind when developing the earthquake event catalogue for our probabilistic seismic

hazard analysis.

In the second of the CDMG publications (Cao et al. 1996), the seismologists developed

estimates o f background seismicity in Southern California, producing a ground motion map for

10% probability o f exceedance in fifty years for the area from background events with magnitude

between 5.0 and 6.5. A distinction between random (aleatory) and modeling (epistemic)

uncertainties is made, noting that parameters used in the recurrence relationship (b-value)

incorporate random uncertainties, while the choice of lower (m0) and upper (m) magnitude

events introduces modeling uncertainties into the analysis. In the end, they present the hazard

map, again utilizing a Monte Carlo approach to estimating the uncertainty associated with the

background seismicity o f Southern California.

In the final paper of this series of publications (Cramer et al. 1996), those involved in the

analysis used a logic tree approach coupled with Monte Carlo simulation techniques to perform

the uncertainty analysis of the seismic hazard in Southern California. They considered nine

separate parameters in the analysis, three having a discrete uncertainty distribution and six having

a continuous distribution. The continuous distributions are assumed to be normal and the discrete

distributions assume equal weights among all the outcomes. The parameters include fault length,21


fault width, fault slip rate, attenuation relationship, magnitude-frequency distribution,

incorporation of blind faults, maximum magnitude, b-value, and the shear modulus o f the earths

crust. From the final analysis, the seismologists at CDMG assert that the maximum magnitude,

the choice of attenuation relationship, the magnitude-frequency distribution, and the slip rate are

the most influential in estimating the uncertainty in seismic hazard.

With an understanding of the sensitivity of certain parameters in a PSHA, an in-depth

look into the process itself is needed.

2.2.2 Probabilistic Seism ic Hazard Analysis

The beginnings of Probabilistic Seismic Hazard Analysis (PSHA) are generally attributed

to C. Allin Cornell (Cornell 1968). PSHA is an analytical methodology that determines the

likelihood that a specific level o f earthquake-induced ground motion will be exceeded at a given

location during a future time period (Equation 2.1).

jV mu aoN{Z)= A-i ! f f i ( m)fi(r )P (Z > z\m,r)drdm (2.1)

'=1 mQ r=0

In equation 2.1, N(z) designates the expected number o f times ground motion exceeds

level Z during time period, /; At is the mean rate of occurrence o f earthquakes between lower and

upper bound magnitudes being considered in the i h source; f(m ) is the probability density

distribution of magnitude within source/; and f(r) is the probability density distribution o f source

distance between the various locations within source / and the site for which the hazard is being

estimated. Finally, P(Z > z | m, r) is the probability that a given earthquake o f magnitude m and

epicentral distance r will exceed ground motion level z.

PSHA is fundamentally different from Deterministic Seismic Hazard Analysis (DSHA)

in that it carries units of time (Hanks and Cornell 1994). And, the appropriateness of PSHA

22


versus DSHA has been debated for the last thirty years. Some seismologists believe that in

determining the mean rates of occurrence o f certain magnitude events from a seismic source, the

uncertainties are so great that it is best to perform a DSHA. They believe that with estimates of

ground shaking for a certain magnitude event from a certain distance, no units o f time (i.e. annual

probability o f recurrence) need to be carried in order to get a picture o f the likely hazard. The

design earthquake or maximum likely earthquake should be the focus o f an earthquake hazard

evaluation, especially in siting hazardous facilities (eg. nuclear power plants).

The steps involved in a probabilistic seismic hazard analysis have been classified

different ways, but one way is to describe the process in four steps. These include: seismic source

determination, recurrence relationship for sources, estimation of ground motion at a site due to

seismic sources, and probability of exceedance o f a certain level of ground motion for the site

(Figure 2 after Reiter 1990).

23


A t e iSauice

S u p L

SOURCES

SIisjsdZbX

Step 2 RECURRENCE

^ ^ U ncertainly in AUetma&ian

Distance

Step 3 GROUND MOTION

Step 4PROBABILITY OF EXCEEDANCE

Figure 2.1 Probabilistic Seismic Hazard Analysis

2.2.2.1 Seismic Source Determination

In seismic source determination, all known faults and tectonic regions are catalogued and

a seismic source model is defined. This model is a combination of faults, represented as line or

plane sources, and area sources. Their locations and geometries are identified and the seismicity

associated with each source is determined. This step usually involves the aggregation of historical

earthquake occurrence data and it is sorted according to seismic source zone. It is a rather

laborious and subjective operation. The less amount o f historical accounts available, the more

24


subjective the source determination. Whenever possible, though, paleoseismic data {i.e. the

location, timing, and size of prehistoric earthquakes) is used to establish seismic sources.

The problems associated with source modeling arise from the fact that for many regions,

there are errors in epicentral locations, especially with older earthquake events. This is primarily

due to the accuracy, or rather inaccuracy, o f older seismic instruments. Before the establishment

of the National Earthquake Information Center (NEIC) as a part o f the United States Geological

Survey (USGS) in the early 1970s, the collection of seismological data from seismographs

located across the country and around the world was not well coordinated. Today, the NEIC

collects seismological data on a 24-hour-a-day basis, monitoring events of body wave magnitude

or mb > 2.5 in the United States (For types of Earthquake Measures, see Appendix 8.1).

As already mentioned, in 1997, the SSHAC issued a report on the use of expert opinion

and uncertainty in PSHA. In this report, as with other reports on seismicity in the continental

United States (Peterson et al. 1996b; Frankel et al. 1996; NIBS 1997), there is a distinction made

between earthquake activity in the Western United States (WUS) and the Eastern United States

(EUS). In the WUS, line sources or faults are the primary seismic source type, and in the EUS,

area sources are the principal seismic source types.

If the seismic source is a fault line or plane, as in the WUS, a number of parameters may

be used to define the seismicity of the source. These include the length, width, endpoints, and dip

angle o f the fault to define its orientation and the style, slip rate, and average displacement on the

fault (for characteristic faults) to define fault movement. All o f the parameters are based on

paleoseismic data, historical earthquake records, and/or expert opinion. Paleoseismic data is the

most reliable source of information, and as previously noted, it is used to define the fault source

parameters whenever possible. Additionally, data from satellite measurements of the earths

surface are now influencing choices o f creep (slip) rates along established fault lines (Figure 2.2).25


With these parameters, then, the maximum earthquake magnitude can be defined for each

fault source. The maximum magnitude is very important to the seismic hazard calculation

(Cramer et al. 1996). In general, estimating maximum magnitudes can be accomplished through

empirical relationships from analysis of historical data, slip rate data, or geologic and geodetic

data. Past research relates fault rupture parameters to maximum earthquake magnitudes. These

parameters include rupture length and area, maximum surface displacement, and average surface

displacement (eg: Bonilla etal. 1984; Wells and Coppersmith 1994).

Historical Expej t PaleoseismicOpinionEarthquake

Records 'Data Geologic and

Geodetic DataSatelliteImaging

Area SourceFault Source

MaximumMagnitude

Mm.X

Figure 2.2 Seismic Source Determination

The most popular measure of maximum magnitude is the maximum moment magnitude

earthquake (Mw or simply M). It is the maximum earthquake that can occur on the fault plane,

based on measurement of the seismic moment, M0 The seismic moment is defined as the seismic

release on a fault, given in Equation 2.2.

M 0 =\ i -~DA (2.2)

The parameter fj, is the modulus of rigidity o f the earths crust (normally taken as 3 x 10"

dyne/cm2s); D is the average displacement on the fault; and A is the rupture area (i.e. L x IV or

26


Rupture Length times Width of the fault). Using Equation 2.2, the moment magnitude is

generated using Hanks and Kanamori (1979) in Equation 2.3:

M = y lo g io M q - 10.7 (2.3)

Alternatively, if the seismic source is an area source, things become more complicated.

Area sources can either be a concentrated zone with a definitive boundary, a regional source with

a fuzzy boundary, or a background zone, designating changes in the character of seismicity

between zones (Budnitz et al. 1997). Area sources were developed for use in a PSHA in the

eastern United States although area sources do exist in the WUS to a limited degree. They are

simplified representations o f the tectonic history o f a region that is not readily comprehended or

accepted by the experts. Subsequently, parameters used to define a seismic area source are not

easily listed, as in the case of a seismic fault source.

Area sources and their boundaries are based on historical earthquake records, recorded

changes in regional stress, and expert opinion (Figure 2.2). In determining the maximum

earthquake magnitude generated from an area source in the EUS, the historical seismicity record

is very important. This record, along with analogies to similar seismic sources, is the primary

basis for the maximum magnitude event. In contrast to a fault source, where the determination of

Mmax is more rigorous {i.e. some estimated empirical relationships do exist), the determination

of the maximum earthquake magnitude for an area source is more an art than a science

(Budnitz et al. 1997).

2.2.2.2 Recurrence Relationship

Once the seismic source model is generated, with an estimation o f the maximum

magnitude expected from each source, the next step is the establishment o f the recurrence of

earthquakes for each source based on the given data. Earthquake recurrence relationships estimate

27


the frequency of occurrence o f magnitude events up to the maximum magnitude. Various

methods for estimating recurrence relationships o f earthquake events exist, and they are typically

different for fault sources and area sources.

For fault sources, recurrence relationships are generally established from historical

earthquake records and geologic data (Figure 2.3). Historical records are used to establish a

recurrence curve for smaller magnitude events, while geologic data is used to estimate the

recurrence of larger events (i.e. characteristic events). In general, the slip rate and the mean return

period are two important parameters that are used to generate a recurrence relationship. The slip

rate, in millimeters per year, is the rate at which each side o f the fault plane moves relative to the

other. In this way, fault movement is modeled as a continual process and the rate at which the

fault displaces relates how quickly stress is building up in the fault.

The mean return period, in years, is the average recurrence rate or interval o f earthquake

events, based on a certain magnitude event. When performing a PSHA, mean return periods need

to be generated for all magnitudes considered in the analysis (i.e. f j (m) in Equation 2.1).

Establishing a recurrence relationship is typically done using one or more of competing models:

the Gutenberg-Richter relationship, the bounded exponential model, or the characteristic

earthquake model (Youngs and Coppersmith 1985).

The most popular recurrence model is the Gutenberg-Richter magnitude-distribution

relationship. In the mid-I950s, Gutenberg and Richter discovered that the number o f earthquake

events occurring in a region is often log-linearly related to the magnitude of earthquakes (i.e.

f ( m ) ~ I0~bm). The unbounded form of the equation (Richter 1958) is given in Equation 2.4.

N(m) = \0a'bM (2.4)

28


In this equation, for a given time period, N(m) is the cumulative number o f earthquakes

of magnitude m or greater, where M designates moment magnitude, and a and b are constants.

The parameter a describes the severity of seismic activity; large values for a indicate a large

number of events over time. The parameter b (i.e. Richter b-value as mentioned in Section 2.2.1)

describes the relative frequency of smaller events to larger ones; large values for b imply that

small earthquakes occur much more frequently than large ones. Using historical earthquake data,

methods for determining the values for a and b include the method of least squares fit or the

maximum likelihood method (Weichert 1980).

Alternatively, the bounded, or truncated, exponential model explicitly assumes a

maximum magnitude event in its calculation. In this model, the recurrence relationship considers

an upper bound on the magnitude, m, as well as a lower bound, m0- The common assumption is

that earthquake events occur on a fault source within a specified time period at a relative

frequency, f ( m ) , which is of the form:

/ (m) - for mQ < m < mu (2.5)

One form of the model, first proposed by Cornell and Van Marke (1969), is a truncated,

shifted exponential distribution, which is renormalized (Equation 2.6). The cumulative number of

earthquakes of magnitude m or greater, N(m), in a given time period is limited between mu and

ma.

- P ( m - m 0 ) _ - P ( m u - m 0 )N{m) = N{m0 )-------------- - (2.6)

| _ e P \ m u ~ m o )

In contrast to the exponential recurrence relationship, the characteristic earthquake model

assumes a characteristic or same-size earthquake frequency density, usually banded at or near

the maximum earthquake magnitude, mu. Historically, this type of model is used less frequently,

29


but it is actually more appropriate to use for certain individual faults, particularly in California

(Youngs and Coppersmith 1985). Furthermore, recent work done in estimating ground shaking

probabilities uses a combination of the two recurrence models for earthquake magnitude-

frequency distributions of certain fault segments (Peterson et al. 1996b).

HistoricalEarthquake

RecordsGeologicData

ExpertOpinion


Recurrencef(m )

Figure 2 3 Earthquake Recurrence

As in the case o f determining maximum magnitude events for seismic area sources, the

recurrence relationships derived for area sources are complicated. They rely heavily on historical

earthquake data and expert opinion (Figure 2.3). With an expert reviewing these historical

records, seismicity rates can be derived for each magnitude level of the area sources. In the

SSHAC report (Budnitz et al. 1997), it is suggested that a truncated exponential magnitude-

distribution relationship is valid, and the values for a and b should be determined using the

maximum likelihood method.

2.2.2.3 Estimation o f Ground Motion

Now, once the recurrence relationship is established for the various sources, a ground

motion attenuation function is used to describe the decay o f seismic waves from the source with

distance for a given earthquake magnitude. Seismic waves decrease in amplitude and change

frequency content as they travel away from their source due to various geological reasons {eg.30


soil type, existence o f land basins, etc). With this geological information, the ground motion as a

function of the distance from the earthquake source and the magnitude o f the event is calculated

( f i ( m )and f j ( r ) in Equation 2.1). But, different ground motion measures can be utilized,

including peak ground acceleration {PGA), peak ground velocity {PGV) and peak ground

displacement (PGD), or spectral acceleration (SA), spectral velocity (.SV), and spectral

displacement (SD). Notably, SA, Sy, and Sn are simply related to each other in terms of the period,

T, or frequency, at, o f a single-degree-of-freedom system (For more details, see Section 3.2.1.1

and Equations 3.7 and 3.8).

While PSHA studies can characterize ground motion at a site using a few different

measures, the most popular method is peak ground acceleration with a few response spectra

ordinates. PGA is defined as the maximum absolute magnitude o f a ground acceleration time

series, while a response spectrum describes the maximum response of a building as a function of

period for a given level o f damping. The SSHAC report (Budnitz et al. 1997) debates the

appropriateness of using PGA, as well as ordinates o f response spectra. They note that the

response spectrum has been accepted as the standard in defining earthquake motions, citing the

Uniform Building Codes use o f standardized response spectra shapes for structural design. But,

they do state the following:

It is recommended that the representation of seismic hazard as a function of

structural frequency be obtained directly through attenuation functions (or equal

formulations) that predict spectral acceleration as a function of structural

frequency, rather than using a fixed spectral shape anchored to a value of PG A

Ground motion is characterized by the magnitude of the earthquake event, the distance

from source to site, and the local site characteristics (Equation 2.7 and Figure 2.4). In this

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equation, z is the ground motion measure (eg. PGA or SA), m is the magnitude, r designates the

distance from source to site, and s is a local soil condition parameter.

z = f ( m , r , s ) (2.7)

As in the case o f determining seismic sources and their recurrence relationships, a

distinction is made between ground motion prediction in the WUS (which contains primarily fault

sources) and the EUS (which contains primarily area sources). The magnitude measures used in

historical earthquake records are different in each region. Historically, the earthquake magnitude

scale used in the WUS is the moment magnitude, M, while the common recorded earthquake

magnitude scale in the EUS is body wave magnitude, mb (For comparison o f Earthquake

Measures, see Appendix 8.1).

Distance Local Soil Magnitude (Source to Site) Conditions


Ground Motionz

Figure 2.4 Ground Motion Estimation

For the distance from source to site, the distinction between the hypocentral (focal)

distance and the epicentral distance is important (Figure 2.5). The hypocentral distance is the

distance from the site to the hypocenter o f the earthquake, the actual point of energy release

below the earths surface. The epicentral distance is the distance from the site to the epicenter, the

point vertically above the hypocenter. Clearly, the epicenter is the projection o f the hypocenter on

the ground surface.

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