probabilistic seismic hazard analysis problem

Probabilistic Seismic Probabilistic Seismic Hazard Analysis Hazard Analysis

ProblemProblem

Hazard AnalysisHazard AnalysisDefinitions: Definitions:

Hazard Hazard –– anticipated ground motion at siteanticipated ground motion at siteRisk Risk –– damage or losses to societydamage or losses to societyProbabilisiticProbabilisitic –– considers uncertainty in input parameters (e.g., considers uncertainty in input parameters (e.g., magnitude, ground motion)magnitude, ground motion)Uncertainty Uncertainty –– aleatory (random), epistemic (variability in mean aleatory (random), epistemic (variability in mean values for different modelsvalues for different modelsDeterministic Deterministic –– considers average level for scenarioconsiders average level for scenarioHazard level Hazard level –– probability level used for building designprobability level used for building designDisaggregationDisaggregation –– magnitude and distance that contribute mostmagnitude and distance that contribute mostMagnitudeMagnitude--frequency distribution frequency distribution –– sizes and rate of sizes and rate of earthquakesearthquakesTimeTime--dependence dependence –– probability increases with time since last probability increases with time since last earthquakeearthquake

Cumulative Normal distribution at two standard deviations

Cumulative Normal distribution at one standard deviation

median +1 sd-2 sd

100%68%

-1 sd +2 sd

84%16%

98%2%

1.8 m

1.9 m1.7 m

Normal DistributionNormal Distribution

Area under curve is Area under curve is probabilityprobability

Normal distribution

Ex. Height: median = 1.8msd=0.1m

median +1 sd-2 sd -1 sd +2 sd

HOWEVER GROUND MOTION FOLLOWS A LOGNORMAL DISTRIBUTION

medianmean

0.2 g

-1.6 -0.9 -0.2-2.3-3.0ln (0.05g) ln(0.1g) ln(0.2g) ln(0.4g) ln(0.8g)

Add 0.7 to ln(median)to get the +1sd

Standard deviation isfactor of two

mode

Pga

Pga

0.4g

Lognormal DistributionLognormal Distribution

Seismic hazard analysis has two Seismic hazard analysis has two model components:model components:

(1)Earthquake-Rupture Forecast (ERF)

Probability of all possible fault-rupture events (M≥~5) for region & time span

(2) Intensity-Measure Relationship (IMR)

Gives Prob(IMT≥IML) for a given site and fault-rupture event

Attenuation Relationships(traditional)(no physics)

Full-Waveform Modeling

(developmental)(more physics)

ρÝ Ý u i = τ ij, j + fi

τ ij = λδijεpp + 2μεij

a bccEarthquake Sources

Ground motion

d1

d2

d3

d4

r1

r2

r3

San Andreas fault

high seismicityzone

peak ground acceleration (pga)

Hazard curve

annu

al ra

te o

f ex

ceed

ing

pga

0.25g

M 7.6

distancepeak

gro

und

acce

lera

tion

M7.6

0.5g

To calculate the hazard curve (annual rate of exceeding ground mTo calculate the hazard curve (annual rate of exceeding ground motions) we:otions) we:

1. Determine magnitude, and distance, and rate of earthquake1. Determine magnitude, and distance, and rate of earthquake

2. Calculate ground motion distribution for that m and d.2. Calculate ground motion distribution for that m and d.

3. Calculate the product:3. Calculate the product:

annual rate of earthquake *probability that earthquake will exceannual rate of earthquake *probability that earthquake will exceed certain ground motion leveled certain ground motion level

4. Sum these rates for all earthquakes in the model at each grou4. Sum these rates for all earthquakes in the model at each ground motion to getnd motion to get

a hazard curve. This curve shows the rate of exceedance of a hazard curve. This curve shows the rate of exceedance of each ground motion.each ground motion.

PROBABILISTIC SEISMIC HAZARD METHODOLOGYPROBABILISTIC SEISMIC HAZARD METHODOLOGY

M7.6 every 250yrsM7.6 every 250yrs-- 0.004 event/yr0.004 event/yr

Probabilistic Seismic HazardProbabilistic Seismic Hazard

Annual rate of exceedance for a given ground motion, zAnnual rate of exceedance for a given ground motion, z

Sources (already discussed)Sources (already discussed)Normal distributionNormal distributionLognormal distributionLognormal distributionMultiplying source rates and probability of exceeding Multiplying source rates and probability of exceeding ground motion levelground motion levelAdding exceedances for a ground motion level to get total Adding exceedances for a ground motion level to get total hazard curvehazard curveApplying Poisson relation to get 10% probability of Applying Poisson relation to get 10% probability of exceedance in 50 yearsexceedance in 50 years

( ) ( ) ( ) ( )1 0

| ,u

o

m rN

i i ii m r

E z f m f r P Z z m r dmdrα=∞

= =

= >∑ ∫ ∫

Earthquake Earthquake sourcesource

Ground Ground motionmotion

15 km

0.2 g

M 6.5, 15 km

PGA

freq

uenc

y

median

Fault 1M 6.5, 15 kmRecurrence: 50 yearsAnnual rate: 0.02

Example 1Example 1

What is the annual rate that ground motion will What is the annual rate that ground motion will exceed exceed 0.05 g0.05 g pga given M 6.5 earthquake occurs?pga given M 6.5 earthquake occurs?

Almost all earthquakes (98%) of M 6.5 at a distance Almost all earthquakes (98%) of M 6.5 at a distance of 15 km will cause ground motion that will exceed of 15 km will cause ground motion that will exceed 0.05g.0.05g.

Annual Rate of ExceedanceAnnual Rate of Exceedance

0.2

M 6.5, 15 km

0.1 0.3 0.4 PGA

frequency

median0.05 g

2%

98%


Recall the equation for the annual rate of Recall the equation for the annual rate of exceedance and simplify for a single source, exceedance and simplify for a single source, distance and magnitude:distance and magnitude:

Ground motion:Ground motion: 0.05g0.05g

Rate of 0.05 g exceedance:Rate of 0.05 g exceedance: 0.020.02

( ) ( )| ,E z P Z z m rα= >

( )0.05 0.02*0.98 0.02E = =


50% of earthquakes of M 6.5 at a distance of 15 km 50% of earthquakes of M 6.5 at a distance of 15 km will cause ground motion that will exceed 0.2g.will cause ground motion that will exceed 0.2g.


0.2

M 6.5, 15 km

0.1 0.3 0.4 PGA

frequency

median0.05 g

50%

50%



Ground motion:Ground motion: 0.05g 0.2g0.05g 0.2g

Rate of 0.05 g exceedance:Rate of 0.05 g exceedance: 0.02 0.010.02 0.01

( ) ( )| ,E z P Z z m rα= >

( )0.2 0.02*0.50 0.01E = =


16% of earthquakes of M 6.5 at a distance of 15 km 16% of earthquakes of M 6.5 at a distance of 15 km will cause ground motion that will exceed 0.4g.will cause ground motion that will exceed 0.4g.


0.2

M 6.5, 15 km

0.1 0.3 0.4 PGA

frequency

median0.05 g

84%

16%



Ground motion:Ground motion: 0.05g 0.2g 0.4g0.05g 0.2g 0.4g

Rate of 0.05 g exceedance:Rate of 0.05 g exceedance: 0.02 0.01 0.0030.02 0.01 0.003

( ) ( )| ,E z P Z z m rα= >

( )0.4 0.02*0.16 0.003E = =

Ann

ual r

ate

grou

nd m

otio

n is

ex

ceed

ed

PGA0.001

0.01

0.1

0.1 g 0.2 g 0.3 g 0.4 g

10% probability of exceedance in 50 years

Hazard CurveHazard Curve

Ground motion:Ground motion: 0.05g0.05g 0.2g0.2g 0.4g0.4gRate of Ground motion exceedance:Rate of Ground motion exceedance: 0.020.02 0.010.01 0.0030.003

0.5 g

15 km

0.4 g

M 7.9, 20 km

PGA

freq

uenc

y

median

Fault 1M 6.5, 15 kmRecurrence: 50 yearsAnnual rate: 0.02

Example 2Example 2

20 km

Fault 2M 7.9, 20 kmRecurrence: 250 yearsAnnual Rate: 0.004

Earthquake Earthquake sourcessources

Ground Ground motionmotion

What is the annual rate that ground motion will What is the annual rate that ground motion will exceed exceed 0.05 g0.05 g pga given M 6.5 and M 7.9 pga given M 6.5 and M 7.9 earthquakes occur?earthquakes occur?

Almost all earthquakes (99.9%) of M 7.9 at a Almost all earthquakes (99.9%) of M 7.9 at a distance of 20 km will cause ground motion that distance of 20 km will cause ground motion that will exceed 0.05g.will exceed 0.05g.


0.4

M 7.9, 20 km

0.2 0.6 0.8 PGA

frequency

median0.05 g

0.1%99.9%



Ground motion:Ground motion: 0.05g0.05g

Rate of 0.05 g exceedance:Rate of 0.05 g exceedance: 0.0240.024

( ) ( )2

1| ,i

iE z P Z z m rα

=

= >∑( )0.05 0.02*0.98 0.004*0.999 0.024E = + =

What is the annual rate that ground motion will What is the annual rate that ground motion will exceed exceed 0.2 g0.2 g pga given M 6.5 and M 7.9 earthquake pga given M 6.5 and M 7.9 earthquake occur?occur?

84% of M 7.9 at a distance of 20 km will cause 84% of M 7.9 at a distance of 20 km will cause ground motion that will exceed 0.2g.ground motion that will exceed 0.2g.


0.4

M 7.9, 20 km

0.2 0.6 0.8 PGA

frequency

median0.05 g

16%84%


Recall the equation for the annual rate of Recall the equation for the annual rate of exceedance:exceedance:

Ground motion:Ground motion: 0.05g 0.2g0.05g 0.2g

Rate of 0.05 g exceedance:Rate of 0.05 g exceedance: 0.024 0.0130.024 0.013

( ) ( )2

1| ,i

iE z P Z z m rα

=

= >∑( )0.2 0.02*0.50 0.004*0.84 0.013E = + =

What is the annual rate that ground motion will What is the annual rate that ground motion will exceed exceed 0.4 g0.4 g pga given M 6.5 and M 7.9 earthquake pga given M 6.5 and M 7.9 earthquake occur?occur?

50% of M 7.9 at a distance of 20 km will cause 50% of M 7.9 at a distance of 20 km will cause ground motion that will exceed 0.4g.ground motion that will exceed 0.4g.


0.4

M 7.9, 20 km

0.2 0.6 0.8 PGA

frequency

median0.05 g

50%50%


Recall the equation for the annual rate of Recall the equation for the annual rate of exceedance:exceedance:

Ground motion:Ground motion: 0.05g 0.2g 0.4g0.05g 0.2g 0.4g

Rate of 0.05 g exceedance:Rate of 0.05 g exceedance: 0.024 0.013 0.0050.024 0.013 0.005

( ) ( )2

1| ,i

iE z P Z z m rα

=

= >∑( )0.2 0.02*0.16 0.004*0.50 0.005E = + =

Ground motion:Ground motion: 0.05g0.05g 0.2g0.2g 0.4g0.4gFault 1:Fault 1: 0.020.02 0.010.01 0.0030.003Fault 2:Fault 2: 0.0040.004 0.0030.003 0.0020.002Rate of Ground motion exceedance:Rate of Ground motion exceedance: 0.0240.024 0.0130.013 0.0050.005

Ann

ual R

ate

grou

nd m

otio

n is

ex

ceed

ed

PGA0.001

0.01

0.1 g 0.2 g 0.3 g 0.4 g

10% probability of exceedance in 50 years

Hazard CurveHazard Curve

0.5 g

HAZARD PRODUCTSHAZARD PRODUCTS http://http://eqhazmaps.usgs.goveqhazmaps.usgs.gov

2002 Hazard Maps - PGA-rock2% probability of exceedance in 50-years

Purple-0.3g and greaterYellow -0.1g and greater

Deaggregations

probabilistic seismic hazard analysis problem

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