Probabilistic Seismic Hazard Analysis

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisOverview

History1969 - Allin Cornell BSSA paperRapid development since that time

Deterministic (DSHA)Assumes a single “scenario”

Select a single magnitude, MSelect a single distance, RAssume effects due to M, R

Probabilistic (PSHA)Assumes many scenarios

Consider all magnitudesConsider all distancesConsider all effects

Ground motion

parameters

Ground motion

parameters

Probabilistic (PSHA)Assumes many scenarios

Consider all magnitudesConsider all distancesConsider all effects

Ground motion

parameters

Why? Because we don’t know when earthquakes will occur, we don’t know where they will occur, and we don’t know how big they will be

Consists of four primary steps:

1. Identification and characterization of all sources

2. Characterization of seismicity of each source

3. Determination of motions from each source

4. Probabilistic calculations

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard Analysis

PSHA characterizes uncertainty in location, size, frequency, and effects of earthquakes, and combines all of them to compute probabilities of different levels of ground shaking

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisUncertainty in source-site distance

Need to specify distance measureBased on distance measure in attenuation relationship

Seismogenicdepth

Vertical Faults

Need to specify distance measureBased on distance measure in attenuation relationship

Dipping Faults

rseis & rrup

Where on fault is rupture most likely to occur?

Source-site distance depends on where rupture occurs

Where is rupture most likely to occur? We don’t know

Source-site distance depends on where rupture occurs

Approach:

rmin rmax

Assume equal likelihood at any pointCharacterize uncertainty probabilistically

pdf for source-site distance

Two practical ways to determine fR(r)

Draw series of concentric circles with equal radius increment

Measure length of fault, Li, between each pair of adjacent circles

Assign weight equal to Li/L to each corresponding distance

Two practical ways to determine fR(r)

Divide entire fault into equal length segments

Compute distance from site to center of each segment

Create histogram of source-site distance. Accuracy increases with increasing number of segments

Linear source

Areal Source

Divide source into equal area elements

Compute distance from center of each element

Create histogram of source-site distance

Divide source into equal volume elements

Compute distance from center of each element

Create histogram of source-site distance

Unequal element areas?

Create histogram using weighting factors - weight according to fraction of total source area

Quick visualization of pdf?

Use concentric circle approach - lets you “see” basic shape of pdf quickly

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisCharacterization of maximum magnitude

Determination of Mmax - same as for DSHA

Empirical correlationsRupture length correlationsRupture area correlationsMaximum surface displacement correlations

“Theoretical” determinationSlip rate correlations

Also need to know distribution of magnitudes

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudes

Given source can produce different earthquakesLow magnitude - oftenLarge magnitude - rare

Gutenberg-RichterSouthern California earthquake data - many faultsCounted number of earthquakes exceeding different magnitude levels over period of many years

log NM

Mean annual rateof exceedance

M = NM / T

Return period(recurrence interval)

TR = 1 /M

0.001 1000 yrs

log TR

0.01 100 yrs

Gutenberg-RichterRecurrence Law

logM = a - bM log TR0

Gutenberg-Richter Recurrence Law

log M = a - bM

Implies that earthquake magnitudes are exponentially distributed (exponential pdf)

Can also be written as

ln M = - M

M = 10a - bM = exp[ - M]

where = 2.303a and = 2.303b.

For an exponential distribution,

fM(m) = e-m

Neglecting events below minimum magnitude, mo

m = exp[ - (m - mo)] m > mo

where = exp[ - mo].

fM(m) = e-m-mo)

For worldwide data (Circumpacific belt),

log m = 7.93 - 0.96M

M = 6 m = 148 /yr TR = 0.0067 yr

M = 7 m = 16.2 TR = 0.062

M = 8 m = 1.78 TR = 0.562

M = 12 m = 0.437 TR = 2.29M > 12 every two years?

Every source has some maximum magnitude

Distribution must be modified to account for Mmax

Bounded G-R recurrence law

)](exp[1)](exp[)](exp[

mmmmmm

Every source has some maximum magnitude

Distribution must be modified to account for Mmax

Bounded G-R recurrence law

)](exp[1)](exp[)](exp[

mmmmmm

log m Bounded G-RRecurrence Law

Characteristic Earthquake Recurrence Law

Paleoseismic investigationsShow similar displacements in each earthquakeInividual faults produce characteristic earthquakesCharacteristic earthquake occur at or near Mmax

Could be caused by geologic constraintsMore research, field observations needed

Seismicity data

Geologic data

CharacteristicEarthquakeRecurrence Law

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisPredictive relationships

M = M*

R = R*

ln YY = Y*

P[Y > Y*| M=M*, R=R*]

Standard error - use to evaluate conditional probability

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisPredictive relationships

M = M*

R = R*

ln YY = Y*

P[Y > Y*| M=M*, R=R*]

Standard error - use to evaluate conditional probability

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisTemporal uncertainty

Poisson process - describes number of occurrences of an event during a given time interval or spatial region.

1. The number of occurrences in one time interval are independent of the number that occur in any other time interval.2. Probability of occurrence in a very short time interval is proportional to length of interval.3. Probability of more than one occurrence in a very short time interval is negligible.

Poisson process

where n is the number of occurrences and m is the average number of occurrences in the time interval of interest.

Poisson process

Letting = t

!)(][n

etnNPtn

Then][...]3[]2[]1[]0[ nPNPNPNPNP

]0[1 NP

Poisson process

Consider an event that occurs, on average, every 1,000 yrs. What is the probability it will occur at least once in a 100 yr period?

= 1/1000 = 0.001

P = 1 - exp[-(0.001)(100)] = 0.0952

What is the probability it will occur at least once in a 1,000 yr period?

P = 1 - exp[-(0.001)(1000)] = 0.632

Solving for ,

tp)1ln(

Then, the annual rate of exceedance for an event with a 10% probability of exceedance in 50 yrs is

0021.050

)1.01ln(

The corresponding return period is TR = 1/ = 475 yrs.

For 2% in 50 yrs, = 0.000404/yr TR = 2475 yrs

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisSummary of uncertainties

Location

Effects

Timing

P[Y > Y*| M=M*, R=R*]

P = 1 - e-t

Source-site distance pdf

Magnitude pdf

Attenuation relationship including standard error

Poisson model

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisCombining uncertainties - probability computations

P[A] =

P[A] = P[A|B1]P[B1] + P[A|B2]P[B2] + … + P[A|BN]P[BN]

B1 B2B3

P[A B1] +

P[A B2] + … +

P[A BN]

TotalProbabilityTheorem

dxfyYPPyYPyYP X )(]|*[][]|*[*][ XXXX

Applying total probability theorem,

where X is a vector of parameters.

We assume that M and R are the most important parameters and that they are independent. Then,

dmdrrfmfrmyYPyYP RM )()(],|*[*][

Above equation gives the probability that y* will be exceeded if an earthquake occurs. Can convert probability to annual rate of exceedance by multiplying probability by annual rate of occurrence of earthquakes.

dmdrrfmfrmyYPyYP RM )()(],|*[*][

dmdrrfmfrmyYP RMy )()(],|*[*

where = exp[ - mo]

If the site of interest is subjected to shaking from more than one site (say Ns sites), then

dmdrrfmfrmyYP RiMiN

s)()(],|*[

For realistic cases, pdfs for M and R are too complicated to integrate analytically. Therefore, we do it numerically.

Dividing the range of possible magnitudes and distances into NM and NR increments, respectively

rmrfmfrmyYP kRijMikjiN

RMS)()(],|*[

This expression can be written, equivalently, as

][][],|*[111

* rRPmMPrmyYP kjkjiN

][][],|*[111

* rRPmMPrmyYP kjkjiN

What does it mean?

All possible magnitudes are considered - contribution of each is weighted by its probability of occurrence

All sites areconsidered

All possible distances are considered - contribution of each is weighted by its probability of occurrence

All possible effects are considered - each weighted by its conditional probability of occurrence

NM x NR possible combinationsEach produces some probability of exceeding y*Must compute P[Y > y*|M=mj,R=rk] for all mj, rk

m3 mNM

Compute conditional probability for each element on gridEnter in matrix (spreadsheet cell)

ln Y M=m2

ln YY = y*

P[Y > y*| M=m2, R=r2]

P[Y > y*| M=m2, R=r1]

P[Y > y*| M=m2, R=r3]

Combining uncertainties - probability computations

m3 mNM

P[Y > y*| M=m2, R=r1]

P[Y > y*| M=m2, R=r3]

P[Y > y*| M=m2, R=r2]

“Build” hazard by:computing conditional probability for each elementmultiplying conditional probability by P[mj], P[rk], i

Repeat for each source - place values in same cells

m3 mNM

P[Y > y*| M=m2, R=r1]

P[Y > y*| M=m2, R=r3]

P[Y > y*| M=m2, R=r2]

When complete (all cells filled for all sources),

Sum all -values for that value of y* y*

m3 mNM

P[Y > y*| M=m2, R=r1]

P[Y > y*| M=m2, R=r3]

P[Y > y*| M=m2, R=r2]

Choose new value of y*Repeat entire processDevelop pairs of (y*, y*) points Plot

log TRlo

SeismicHazardCurve

log TRlo

Seismic hazard curve shows the mean annual rate of exceedance of a particular ground motion parameter. A seismic hazard curve is the ultimate result of a PSHA.

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisUsing seismic hazard curves

amax=0.30g

log TRlo

Probability of exceeding amax = 0.30g in a 50 yr period?

P = 1 - e-t

= 1 - exp[-(0.001)(50)] = 0.049 = 4.9%

In a 500 yr period?

P = 0.393 = 39.3%

amax=0.21g

log TRlo

0.0021

What peak acceleration has a 10% probability of being exceeded in a 50 yr period?

10% in 50 yrs: = 0.0021or

TR = 475 yrs

Use seismic hazard curve to find amax value corresponding to = 0.0021

Using seismic hazard curves

475 yrs

log TRlo

Contribution of sources

Can break -values down into contributions from each sourcePlot seismic hazard curves for each source and total seismic hazard curve (equal to sum of source curves)Curves may not be parallel, may crossShows which source(s) most important

Can develop seismic hazard curves for different ground motion parameters

Peak accelerationSpectral accelerationsOther

Choose desired -valueRead corresponding parameter values from seismic hazard curves

amax, Sa

log TRlo

Can develop seismic hazard curves for different ground motion parameters

Peak accelerationSpectral accelerationsOther

Choose desired -valueRead corresponding parameter values from seismic hazard curves

amax, Sa

log TRlo

0.0001

Crustal

Intraplate

Interplate

2% in 50 yrs

Peak acceleration

2% in 50 yrs

Sa(T = 3 sec)

0.0001

Crustal

Intraplate

Interplate

Find spectral acceleration values for different periods at constant All Sa values have same -value same probability of exceedance

Uniform hazard spectrum (UHS)

UniformHazardSpectrum

Common question:

What magnitude & distance does that amax value correspond to?

Disaggregation (De-aggregation)

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

100 km

125 km

150 km

175 km

200 km

0.01 0.01 0.02 0.03 0.03 0.02 0.01 0.01

0.030.02 0.04 0.04 0.05 0.04 0.03 0.02

0.000.00 0.00 0.00 0.01 0.00 0.00 0.00

0.030.03 0.05 0.06 0.09 0.06 0.05 0.02

0.030.03 0.05 0.05 0.08 0.05 0.05 0.02

0.020.02 0.03 0.04 0.05 0.03 0.02 0.01

0.010.01 0.02 0.03 0.05 0.02 0.01 0.00

0.000.00 0.01 0.01 0.03 0.01 0.01 0.00

Total hazard includes contributions from all combinations of M & R.

Common question:

What magnitude & distance does that amax value correspond to?

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

100 km

125 km

150 km

175 km

200 km

0.01 0.01 0.02 0.03 0.03 0.02 0.01 0.01

0.030.02 0.04 0.04 0.05 0.04 0.03 0.02

0.000.00 0.00 0.00 0.01 0.00 0.00 0.00

0.030.03 0.05 0.06 0.09 0.06 0.05 0.02

0.030.03 0.05 0.05 0.08 0.05 0.05 0.02

0.020.02 0.03 0.04 0.05 0.03 0.02 0.01

0.010.01 0.02 0.03 0.05 0.02 0.01 0.00

0.000.00 0.01 0.01 0.03 0.01 0.01 0.00

Total hazard includes contributions from all combinations of M & R.

Break hazard down into contributions to “see where hazard is coming from.”

M=7.0 at R=75 km

USGS disaggregations

Seattle, WA

2% in 50 yrs (TR = 2475 yrs)

Sa(T = 0.2 sec)

Olympia, WA

2% in 50 yrs (TR = 2475 yrs)

Sa(T = 0.2 sec)

Olympia, WA

2% in 50 yrs (TR = 2475 yrs)

Sa(T = 1.0 sec)

ln Y M=m2

ln Y Y = y*

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisDisaggregation (De-aggregation)

Another disaggregation parameter

= -1.6

= -0.8

= 1.2 = 2.2

For low y*, most values will be negative

For high y*, most values will be positive and large

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisLogic tree methods

Not all uncertainty can be described by probability distributions

Most appropriate model may not be clear• Attenuation relationship• Magnitude distribution• etc.

Experts may disagree on model parameters• Fault segmentation• Maximum magnitude• etc.

AttenuationModel

MagnitudeDistribution Mmax

BJF(0.5)

A & S(0.5)

G-R(0.7)

Char.(0.3)

G-R(0.7)

Char.(0.3)

7.0 (0.2)7.2 (0.6)7.5 (0.2)

AttenuationModel

BJF(0.5)

A & S(0.5)

G-R(0.7)

Char.(0.3)

G-R(0.7)

Char.(0.3)

7.0 (0.2)7.2 (0.6)7.5 (0.2)

Sum of weighting factors coming out of each node must equal 1.0

AttenuationModel

BJF(0.5)

A & S(0.5)

G-R(0.7)

Char.(0.3)

G-R(0.7)

Char.(0.3)

7.0 (0.2)7.2 (0.6)7.5 (0.2)

0.5x0.7x0.2 = 0.07

Final value of Y is obtained as weighted average of all values given by terminal branches of logic tree

AttenuationModel

BJF(0.5)

A & S(0.5)

G-R(0.7)

Char.(0.3)

G-R(0.7)

Char.(0.3)

7.0 (0.2)7.2 (0.6)7.5 (0.2)

0.070.210.07

0.030.090.03

0.070.210.07

0.030.090.03

Recent PSHA logic tree included:

Cascadia interplate2 attenuation relationships2 updip boundaries3 downdip boundaries2 return periods4 segmentation models2 maximum magnitude approaches

192 terminal branches

Cascadia intraplate2 intraslab geometries3 maximum magnitudes2 a-values2 b-values

Seattle Fault and Puget Sound Fault2 attenuation relationships3 activity states3 maximum magnitudes2 recurrence models2 slip rates

72 terminal branches for Seattle Fault 72 terminal branches for Puget Sound Fault

Crustal areal source zones7 source zones2 attenuation relationships3 maximum magnitudes2 recurrence models3 source depths

Total PSHA required analysis of 612 combinations

Probabilistic Seismic Hazard Analysis

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