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

rhypo

rseis

rrup

rjb

Seismogenicdepth

Vertical Faults


Need to specify distance measureBased on distance measure in attenuation relationship

Dipping Faults

rhypo

rseis

rrup

rjb=0

rhypo

rseis & rrup

rjb


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

r

fR(r)

rmin rmax

Assume equal likelihood at any pointCharacterize uncertainty probabilistically

pdf for source-site distance


Two practical ways to determine fR(r)

rmin

rmax

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)

rmin

rmax

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


M

NM

M

log NM

M

log M


Mean annual rateof exceedance

M = NM / T

M

log M


Return period(recurrence interval)

TR = 1 /M

0.001 1000 yrs

log TR

0.01 100 yrs

M

log M


Gutenberg-RichterRecurrence Law

logM = a - bM log TR0

10a

b


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


Then

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].

Then,

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[

max

max

mmmmmm

o

oom


Every source has some maximum magnitude

Distribution must be modified to account for Mmax

Bounded G-R recurrence law

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

max

max

mmmmmm

o

oom

MMmax

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


MMmax

log m

Seismicity data

Geologic data

CharacteristicEarthquakeRecurrence Law

Probabilistic Seismic Hazard AnalysisProbabilistic Seismic Hazard AnalysisPredictive relationships

MMmax

log m

log R

ln Y

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

log R

ln Y

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

!][

nenNP

n

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

e t1


Poisson process

eP t1

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

Size

Effects

Timing

fR(r)

fM(m)

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

B4B5

A

U

P[A B1] +

U

P[A B2] + … +

U

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

iiy

s)()(],|*[

1*

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

k

N

j

N

iy

RMS)()(],|*[

111*

This expression can be written, equivalently, as

][][],|*[111

* rRPmMPrmyYP kjkjiN

k

N

j

N

iy

RMS


][][],|*[111

* rRPmMPrmyYP kjkjiN

k

N

j

N

iy

RMS

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

m1 m2

r1

m3 mNM

rNR


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

log R

ln Y M=m2

r1

ln YY = y*

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

r2

r3

rN

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

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

Combining uncertainties - probability computations


m1 m2

r1

m3 mNM

rNR

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


m1 m2

r1

m3 mNM

rNR

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*


m1 m2

r1

m3 mNM

rNR

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

y*

log TRlo

g y

*

SeismicHazardCurve


y*

log TRlo

g y

*

amax

log TRlo

g a

max

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

g a

max

0.001

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

g a

max

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


amax

log TRlo

g a

max

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


Total

12

3


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

g a

max

amax

0.1

0.01

0.001

0.0001

Crustal

Intraplate

Interplate


2% in 50 yrs

Peak acceleration


2% in 50 yrs

Sa(T = 3 sec)

amax

0.1

0.01

0.001

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)

Sa

T

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

25 km

75 km

100 km

125 km

150 km

175 km

200 km

50 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

25 km

75 km

100 km

125 km

150 km

175 km

200 km

50 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)

log R

ln Y M=m2

r1

ln Y Y = y*

r2r3

rN

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

Another disaggregation parameter

y

yy

ln

ln*ln

= -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)

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

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

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)

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

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

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)

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

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

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)

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

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

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

w


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

Documents

sourcesite distanceneed

sourcesite distancewhere

sourcesite distanceapproach

dont knowsourcesite

single distance

distance measurebased

single scenarioselect

single magnitude