PSHA

Probabilistic Seismic Hazard AnalysisOverviewHistory1969 - Allin Cornell BSSA paperRapid development since that time

Probabilistic Seismic Hazard AnalysisOverviewDeterministic (DSHA)Assumes a single scenarioSelect a single magnitude, MSelect a single distance, RAssume effects due to M, R

Probabilistic (PSHA)Assumes many scenariosConsider all magnitudesConsider all distancesConsider all effects

Probabilistic Seismic Hazard AnalysisOverview

Probabilistic (PSHA)Assumes many scenariosConsider all magnitudesConsider all distancesConsider all effectsGround motion parametersWhy? Because we dont know when earthquakes will occur, we dont know where they will occur, and we dont know how big they will be

Consists of four primary steps:1. Identification and characterization of all sources2. Characterization of seismicity of each source3. Determination of motions from each source4. Probabilistic calculationsProbabilistic Seismic Hazard AnalysisPSHA 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 AnalysisUncertainty in source-site distanceNeed to specify distance measureBased on distance measure in attenuation relationship

Probabilistic Seismic Hazard AnalysisUncertainty in source-site distanceNeed to specify distance measureBased on distance measure in attenuation relationship

Probabilistic Seismic Hazard AnalysisUncertainty in source-site distanceWhere on fault is rupture most likely to occur?Source-site distance depends on where rupture occurs

Probabilistic Seismic Hazard AnalysisUncertainty in source-site distanceWhere is rupture most likely to occur? We dont knowSource-site distance depends on where rupture occurs

Probabilistic Seismic Hazard AnalysisUncertainty in source-site distanceApproach:rminrmaxrfR(r)rminrmaxAssume equal likelihood at any pointCharacterize uncertainty probabilisticallypdf for source-site distance

Probabilistic Seismic Hazard AnalysisUncertainty in source-site distanceTwo practical ways to determine fR(r)rminrmaxDraw 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

Probabilistic Seismic Hazard AnalysisUncertainty in source-site distanceTwo practical ways to determine fR(r)rminrmaxDivide 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 segmentsLinear source

Probabilistic Seismic Hazard AnalysisUncertainty in source-site distanceAreal SourceDivide source into equal area elements

Compute distance from center of each element

Create histogram of source-site distance

Probabilistic Seismic Hazard AnalysisUncertainty in source-site distanceDivide source into equal volume elements

Compute distance from center of each element

Create histogram of source-site distance

Probabilistic Seismic Hazard AnalysisUncertainty in source-site distanceUnequal element areas?

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

Probabilistic Seismic Hazard AnalysisUncertainty in source-site distanceQuick visualization of pdf?

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

Probabilistic Seismic Hazard AnalysisCharacterization of maximum magnitudeDetermination of Mmax - same as for DSHAEmpirical correlationsRupture length correlationsRupture area correlationsMaximum surface displacement correlationsTheoretical determinationSlip rate correlationsAlso need to know distribution of magnitudes

Probabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesGiven 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

Probabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesMNM

Mlog lMProbabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesMean annual rateof exceedance

lM = NM / T

Mlog lMProbabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesReturn period(recurrence interval)

TR = 1 / lM0.0011000 yrslog TR0.01100 yrs

Mlog lMProbabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesGutenberg-RichterRecurrence Law

log lM = a - bMlog TR010ab

Probabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesGutenberg-Richter Recurrence Law

log lM = a - bM

Implies that earthquake magnitudes are exponentially distributed (exponential pdf)

Can also be written as

ln lM = a - bM

Probabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesThen

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

where a = 2.303a and b = 2.303b.

For an exponential distribution,

fM(m) = b e-b m

Probabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesNeglecting events below minimum magnitude, mo

lm = n exp[a - b(m - mo)]m > mo

where n = exp[a - b mo].

Then,

fM(m) = b e-b (m-mo)

Probabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesFor worldwide data (Circumpacific belt),

log lm = 7.93 - 0.96M

M = 6 lm = 148 /yrTR = 0.0067 yrM = 7 lm = 16.2TR = 0.062M = 8 lm = 1.78TR = 0.562

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

Probabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesEvery source has some maximum magnitudeDistribution must be modified to account for MmaxBounded G-R recurrence law

Probabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesEvery source has some maximum magnitudeDistribution must be modified to account for MmaxBounded G-R recurrence law

MMmaxlog lmBounded G-RRecurrence Law

Probabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesCharacteristic Earthquake Recurrence Law

Paleoseismic investigationsShow similar displacements in each earthquakeInividual faults produce characteristic earthquakesCharacteristic earthquake occur at or near MmaxCould be caused by geologic constraintsMore research, field observations needed

Probabilistic Seismic Hazard AnalysisDistribution of earthquake magnitudesMMmaxlog lmSeismicity dataGeologic dataCharacteristicEarthquakeRecurrence Law

Probabilistic Seismic Hazard AnalysisPredictive relationshipsMMmaxlog lmlog Rln YM = M*R = R*ln YY = Y*P[Y > Y*| M=M*, R=R*]Standard error - use to evaluate conditional probability

Probabilistic Seismic Hazard AnalysisPredictive relationshipsMlog Rln YM = M*R = R*ln YY = Y*P[Y > Y*| M=M*, R=R*]Standard error - use to evaluate conditional probability

Probabilistic Seismic Hazard AnalysisTemporal uncertaintyPoisson 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.

Probabilistic Seismic Hazard AnalysisTemporal uncertaintyPoisson processwhere n is the number of occurrences and m is the average number of occurrences in the time interval of interest.

Probabilistic Seismic Hazard AnalysisTemporal uncertaintyPoisson process

Letting m = ltThen

Probabilistic Seismic Hazard AnalysisTemporal uncertaintyPoisson processConsider 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?

l = 1/1000 = 0.001

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

Probabilistic Seismic Hazard AnalysisTemporal uncertaintyWhat 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 l,

Probabilistic Seismic Hazard AnalysisTemporal uncertaintyThen, the annual rate of exceedance for an event with a 10% probability of exceedance in 50 yrs isThe corresponding return period is TR = 1/l = 475 yrs.

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

Probabilistic Seismic Hazard AnalysisSummary of uncertaintiesLocation

Size

Effects

TimingfR(r)

fM(m)

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

P = 1 - e-ltPoisson model

Probabilistic 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]TotalProbabilityTheorem

Probabilistic Seismic Hazard AnalysisCombining uncertainties - probability computationsApplying 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,

Probabilistic Seismic Hazard AnalysisCombining uncertainties - probability computationsAbove 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.where n = exp[a - bmo]

Probabilistic Seismic Hazard AnalysisCombining uncertainties - probability computationsIf the site of interest is subjected to shaking from more than one site (say Ns sites), thenFor realistic cases, pdfs for M and R are too complicated to integrate analytically. Therefore, we do it numerically.

Probabilistic Seismic Hazard AnalysisCombining uncertainties - probability computationsDividing the range of possible magnitudes and distances into NM and NR increments, respectivelyThis expression can be written, equivalently, as

Probabilistic Seismic Hazard AnalysisCombining uncertainties - probability computationsWhat does it mean?

Probabilistic Seismic Hazard AnalysisCombining uncertainties - probability computationsNM x NR possible combinationsEach produces some probability of exceeding y*Must compute P[Y > y*|M=mj,R=rk] for all mj, rk

Probabilistic Seismic Hazard AnalysisCompute conditional probability for each element on gridEnter in matrix (spreadsheet cell)Combining uncertainties - probability computations

Probabilistic Seismic Hazard AnalysisCombining uncertainties - probability computationsBuild hazard by:computing conditional probability for each elementmultiplying conditional probability by P[mj], P[rk], niRepeat for each source - place values in same cells

Probabilistic Seismic Hazard AnalysisCombining uncertainties - probability computationsWhen complete (all cells filled for all sources),

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

Probabilistic Seismic Hazard AnalysisCombining uncertainties - probability computationsChoose new value of y*Repeat entire processDevelop pairs of (y*, ly*) points Ploty*log TRlog ly*SeismicHazardCurve

Probabilistic Seismic Hazard AnalysisCombining uncertainties - probability computationsy*log TRlog ly*amaxlog TRlog lamaxSeismic 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 AnalysisUsing seismic hazard curvesProbability of exceeding amax = 0.30g in a 50 yr period?

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

In a 500 yr period?

P = 0.393 = 39.3%

Probabilistic Seismic Hazard Analysisamax=0.21glog TRlog lamax0.0021What peak acceleration has a 10% probability of being exceeded in a 50 yr period?

10% in 50 yrs: l = 0.0021orTR = 475 yrs

Use seismic hazard curve to find amax value corresponding to l = 0.0021Using seismic hazard curves475 yrs

Probabilistic Seismic Hazard Analysisamaxlog TRlog lamaxContribution of sources

Can break l-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 importantUsing seismic hazard curvesTotal123

Probabilistic Seismic Hazard AnalysisCan develop seismic hazard curves for different ground motion parametersPeak accelerationSpectral accelerationsOtherChoose desired l-valueRead corresponding parameter values from seismic hazard curvesUsing seismic hazard curves

Probabilistic Seismic Hazard AnalysisCan develop seismic hazard curves for different ground motion parametersPeak accelerationSpectral accelerationsOtherChoose desired l-valueRead corresponding parameter values from seismic hazard curvesUsing seismic hazard curves

lamax0.10.010.0010.0001CrustalIntraplateInterplateProbabilistic Seismic Hazard Analysis2% in 50 yrs

Peak acceleration

Probabilistic Seismic Hazard Analysis2% in 50 yrs

Sa(T = 3 sec)lamax0.10.010.0010.0001CrustalIntraplateInterplate

Probabilistic Seismic Hazard AnalysisFind spectral acceleration values for different periods at constant lAll Sa values have same l-value same probability of exceedanceUniform hazard spectrum (UHS)UniformHazardSpectrum

Probabilistic Seismic Hazard AnalysisCommon question:

What magnitude & distance does that amax value correspond to?Disaggregation (De-aggregation)Total hazard includes contributions from all combinations of M & R.

Probabilistic Seismic Hazard AnalysisCommon question:

What magnitude & distance does that amax value correspond to?Disaggregation (De-aggregation)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

Probabilistic Seismic Hazard AnalysisUSGS disaggregationsDisaggregation (De-aggregation)Seattle, WA

2% in 50 yrs (TR = 2475 yrs)

Sa(T = 0.2 sec)

Probabilistic Seismic Hazard AnalysisUSGS disaggregationsDisaggregation (De-aggregation)Olympia, WA

2% in 50 yrs (TR = 2475 yrs)

Sa(T = 0.2 sec)

Probabilistic Seismic Hazard AnalysisUSGS disaggregationsDisaggregation (De-aggregation)Olympia, WA

2% in 50 yrs (TR = 2475 yrs)

Sa(T = 1.0 sec)

log Rln YM=m2r1ln YY = y*r2r3rNProbabilistic Seismic Hazard AnalysisDisaggregation (De-aggregation)Another disaggregation parametere = -1.6e = -0.8e = 1.2e = 2.2For low y*, most e values will be negative

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

Probabilistic Seismic Hazard AnalysisLogic tree methodsNot 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.

Probabilistic Seismic Hazard AnalysisLogic tree methodsAttenuationModelMagnitudeDistributionMmaxBJF(0.5)A & S(0.5)G-R(0.7)Char.(0.3)G-R(0.7)Char.(0.3)

Probabilistic Seismic Hazard AnalysisLogic tree methodsAttenuationModelMagnitudeDistributionMmaxBJF(0.5)A & S(0.5)G-R(0.7)Char.(0.3)G-R(0.7)Char.(0.3)Sum of weighting factors coming out of each node must equal 1.0

Probabilistic Seismic Hazard AnalysisLogic tree methodsAttenuationModelMagnitudeDistributionMmaxBJF(0.5)A & S(0.5)G-R(0.7)Char.(0.3)G-R(0.7)Char.(0.3)0.5x0.7x0.2 = 0.07

Final value of Y is obtained as weighted average of all values given by terminal branches of logic treeProbabilistic Seismic Hazard AnalysisLogic tree methodsAttenuationModelMagnitudeDistributionMmaxBJF(0.5)A & S(0.5)G-R(0.7)Char.(0.3)G-R(0.7)Char.(0.3)w

Probabilistic Seismic Hazard AnalysisLogic tree methodsRecent PSHA logic tree included:

Cascadia interplate2 attenuation relationships2 updip boundaries3 downdip boundaries2 return periods4 segmentation models2 maximum magnitude approaches 192 terminal branches

Probabilistic Seismic Hazard AnalysisLogic tree methodsRecent PSHA logic tree included:

Cascadia intraplate2 intraslab geometries3 maximum magnitudes2 a-values2 b-values 24 terminal branches

Probabilistic Seismic Hazard AnalysisLogic tree methodsRecent PSHA logic tree included:

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

Probabilistic Seismic Hazard AnalysisLogic tree methodsRecent PSHA logic tree included:

Crustal areal source zones7 source zones2 attenuation relationships3 maximum magnitudes2 recurrence models3 source depths 252 terminal branches

Total PSHA required analysis of 612 combinations