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Seismic Risk of Co-Located Critical 

Facilities – Effects of Correlation and 

Uncertainty

Martin McCann

Stanford University

& 

Jack R. Benjamin & Associates, Inc
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Overview

• Consider the following:

– One or multiple dams are located upstream of a critical facility. 

What is the risk the critical facility may fail as a result of an 

earthquake or as a result of upstream dam failure*

.

• Conducted a parametric study to evaluate the influence of a number of 

sources of dependence/correlation associated with the seismic hazard on 

seismic risk estimates.

• Parametric study for a small group of dams

*

We have to assume of course that upstream dam results in inundation and failure of the 

downstream facility; otherwise the problem would not be a much Interest.
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Taxonomy of Uncertainties
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Sources of Dependence & Correlation

• There are three sources of dependence & correlation 

that are considered.

• These are:

– Epistemic uncertainty in seismic hazard estimates,

– Aleatory uncertainty in the median ground motion,

– Aleatory uncertainty and spatial correlation of ground motions for 

sites located in regional proximity one another.
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Chance of Failure of the Downstream Facility

• The potential for failure of the downstream facility is a 

function of the ground motions that may occur at the 

facility itself and at the upstream dams, during the same 

seismic event. 

For a facility on its own (no upstream dams) the frequency

of failure can be estimate by:

= ∫ a da DS P f DS a F DS ( ) ( | )

,

ν ν

Seismic

Hazard

Seismic

Fragility for 

the Facility
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Chance of Failure of the Downstream Facility

• When the upstream dams come into the picture a 

number of things change

P( fDS | EQ) = [P( fDS | EQ,(a(x))

Now the seismic fragility of the downstream facility can be 

expressed as follows:

+ P( fDS | DF)P(DF | EQ,a(x))P(a(x) | EQ)]

(Original part – previous page)

(New piece, that includes the potential for the dams failing and causing failure)
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Chance of Failure of the Downstream Facility

EQ – Earthquake of a given magnitude, seismic source, and location on that

 source.

a(x) – spatial field of correlated ground motions, given an EQ

The frequency of failure is now:

P f DF P DF EQ a x P a x EQ da x dEQ

P f EQ a x

DS

DS

EQ a x

DS

( | ) ( | , ( )) ( ( ) | ) ( )

[ ( | ,( ( ))

( )

= + ∫ ∫ ν
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Probabilistic Seismic Hazard Analysis

• Modeling epistemic uncertainties is the primary focus of a PSHA.

• The NRC currently requires that a SSHAC Level 3 or 4 be carried 

out for NPP sites.

• PSHAs are carried out to estimate the ground motion at a single 

site; a point in space.

• Logic tree modeling

– Model uncertainties / alternative interpretations of available scientific 

data (seismic source models, ground motion attenuation models, etc.)

– Parametric uncertainties (Mmax, earthquake rates, etc.)

• Expert elicitation; formal & structured process
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Illustration of Logic Tree For Fault Sources
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Typical PSHA Results
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PSHA Results – Discrete Set of Curves

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

0.01 0.1 1 10

PGA (g)

Frequency of Exceedance per Year
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Epistemic Uncertainty in Seismic Hazard Estimates

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

0.01 0.1 1 10

PGA (g)

Annual frequency of exceedance

Clearly the estimate of the 

frequency of failure will 

differ significantly for the 

hazard curves shown.

The difference in the level 

and shape of the hazard 

curves (slope) will lead to 

very different estimates of 

the frequency of failure. 
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Aleatory Uncertainty in Ground Motions

• There is considerable aleatory variability in earthquake ground 

motions.

• Logarithmic standard deviation ~ 0.6 (we have data to +/-2.5-3 

standard deviations).

• Empirical studies have identified 2 basic parts to the aleatory 

uncertainty in ground motions:

– Inter-event term (different between earthquakes of the same 

magnitude)

– Intra-event terms
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Ground Motion Variability

2 2 2

T σ I σ =τ +

τ =

σ I=

Ground motion aleatory variability

where,

Inter-event terms, quantifies the random, but systematic 

difference between ground motions from earthquakes of the 

same magnitude.

Intra-event term, quantifies the random variability of ground 

motions within an event; these motions are spatially 

correlated (based on separation distance of sites).
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Earthquake Ground Motion Variability Tau 

Effect

For earthquakes of the

same magnitude (say M6.5),

the median estimate of ground 

motions can systematically, 

but randomly vary from one 

earthquake to the next.

This (and the next source of 

correlation adds a 

complication to the hazard 

analysis for multiple sites and 

to a risk analysis.
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Ground Motion – Spatial Correlations

• Seismologists have estimated the spatial correlation of 

ground motions based on the separation distance of 

sites

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 20 40 60 80 100

Distance (km)

Correlation Coefficient
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Applications

• Delta Risk Management Study in California – seismic risk for 1,200+ 

miles of levees (dams since water is up against the levees 24-7)

• Insurance portfolios

• Lifeline risk analysis

• Simulation methods are used to estimate the correlated 

ground motions (account for the Tau effect and σI.
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Fault Source, Critical Facility & a Series of 

Upstream Dam

Fault

A series of dams on a river system
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Observations

• The evaluation of downstream risks is very site/situation specific:

– Number of upstream facilities

– The relative location of those facilities (separation distance)

– Magnitude of earthquakes that can occur

– Seismic fragility of the downstream facility and the upstream dams

– Location of seismic sources relative to the various facilities

• Depending on the above factors, the impact can be relatively small (less 

than a factor of 2 in the frequency of failure), to considerably greater 

(>10).

• Risk studies and quantitative parametric studies make it clear that failure 

to account for the sources of correlation is unconservative (risk is underestimated).