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Probabilistic Seismic Hazard Analysis

Probabilistic Seismic Hazard Analysis. Overview. History 1969 - Allin Cornell BSSA paper Rapid development since that time. Probabilistic Seismic Hazard Analysis. Overview. Deterministic (DSHA) Assumes a single “scenario” Select a single magnitude, M Select a single distance, R

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Probabilistic Seismic Hazard Analysis

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  1. Probabilistic Seismic Hazard Analysis Overview • History • 1969 - Allin Cornell BSSA paper • Rapid development since that time

  2. Probabilistic Seismic Hazard Analysis Overview • Deterministic (DSHA) • Assumes a single “scenario” • Select a single magnitude, M • Select a single distance, R • Assume effects due to M, R • Probabilistic (PSHA) • Assumes many scenarios • Consider all magnitudes • Consider all distances • Consider all effects Ground motion parameters Ground motion parameters

  3. Probabilistic Seismic Hazard Analysis Overview • Probabilistic (PSHA) • Assumes many scenarios • Consider all magnitudes • Consider all distances • Consider all effects 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 Ground motion parameters

  4. Probabilistic Seismic Hazard Analysis • 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 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

  5. Vertical Faults rjb rrup Seismogenic depth rseis rhypo Probabilistic Seismic Hazard Analysis Uncertainty in source-site distance Need to specify distance measure Based on distance measure in attenuation relationship

  6. Dipping Faults rjb=0 rjb rseis rseis & rrup rrup rhypo rhypo Probabilistic Seismic Hazard Analysis Uncertainty in source-site distance Need to specify distance measure Based on distance measure in attenuation relationship

  7. Probabilistic Seismic Hazard Analysis Uncertainty in source-site distance Where on fault is rupture most likely to occur? Source-site distance depends on where rupture occurs

  8. Probabilistic Seismic Hazard Analysis Uncertainty in source-site distance Where is rupture most likely to occur? We don’t know Source-site distance depends on where rupture occurs

  9. Probabilistic Seismic Hazard Analysis Uncertainty in source-site distance Approach: Assume equal likelihood at any point Characterize uncertainty probabilistically rmin fR(r) rmax r rmin rmax pdf for source-site distance

  10. Probabilistic Seismic Hazard Analysis Uncertainty in 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 rmin rmax

  11. Probabilistic Seismic Hazard Analysis Uncertainty in source-site 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 rmin rmax Linear source

  12. Probabilistic Seismic Hazard Analysis Uncertainty in source-site distance Divide source into equal area elements Compute distance from center of each element Create histogram of source-site distance Areal Source

  13. Probabilistic Seismic Hazard Analysis Uncertainty in source-site distance Divide source into equal volume elements Compute distance from center of each element Create histogram of source-site distance

  14. Probabilistic Seismic Hazard Analysis Uncertainty in source-site distance Unequal element areas? Create histogram using weighting factors - weight according to fraction of total source area

  15. Probabilistic Seismic Hazard Analysis Uncertainty in source-site distance Quick visualization of pdf? Use concentric circle approach - lets you “see” basic shape of pdf quickly

  16. Probabilistic Seismic Hazard Analysis Characterization of maximum magnitude • Determination of Mmax - same as for DSHA • Empirical correlations • Rupture length correlations • Rupture area correlations • Maximum surface displacement correlations • “Theoretical” determination • Slip rate correlations Also need to know distribution of magnitudes

  17. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes • Given source can produce different earthquakes • Low magnitude - often • Large magnitude - rare • Gutenberg-Richter • Southern California earthquake data - many faults • Counted number of earthquakes exceeding different • magnitude levels over period of many years

  18. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes NM log NM M M

  19. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes log lM Mean annual rate of exceedance lM = NM / T M

  20. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes log lM 100 yrs 0.01 Return period (recurrence interval) TR = 1 / lM 0.001 1000 yrs log TR M

  21. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes log lM 10a b Gutenberg-Richter Recurrence Law log lM = a - bM log TR 0 M

  22. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes • Gutenberg-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

  23. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes • Then • lM = 10a - bM = exp[a - bM] • where a = 2.303a and b = 2.303b. • For an exponential distribution, • fM(m) = b e-b m

  24. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes • Neglecting 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)

  25. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes • For worldwide data (Circumpacific belt), • log lm = 7.93 - 0.96M • M = 6 lm = 148 /yr TR = 0.0067 yr • M = 7 lm = 16.2 TR = 0.062 • M = 8 lm = 1.78 TR = 0.562 • M = 12 lm = 0.437 TR = 2.29 M > 12 every two years?

  26. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes Every source has some maximum magnitude Distribution must be modified to account for Mmax Bounded G-R recurrence law

  27. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes Every source has some maximum magnitude Distribution must be modified to account for Mmax Bounded G-R recurrence law log lm Bounded G-R Recurrence Law Mmax M

  28. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes • Characteristic Earthquake Recurrence Law • Paleoseismic investigations • Show similar displacements in each earthquake • Inividual faults produce characteristic earthquakes • Characteristic earthquake occur at or near Mmax • Could be caused by geologic constraints • More research, field observations needed

  29. Probabilistic Seismic Hazard Analysis Distribution of earthquake magnitudes log lm Characteristic Earthquake Recurrence Law Seismicity data Geologic data Mmax M

  30. Probabilistic Seismic Hazard Analysis Predictive relationships Standard error - use to evaluate conditional probability log lm P[Y > Y*| M=M*, R=R*] ln Y Y = Y* ln Y M = M* R = R* log R Mmax M

  31. Probabilistic Seismic Hazard Analysis Predictive relationships Standard error - use to evaluate conditional probability ln Y P[Y > Y*| M=M*, R=R*] ln Y Y = Y* M = M* R = R* log R M

  32. Probabilistic Seismic Hazard Analysis Temporal 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.

  33. Probabilistic Seismic Hazard Analysis Temporal uncertainty Poisson process where n is the number of occurrences and m is the average number of occurrences in the time interval of interest.

  34. Probabilistic Seismic Hazard Analysis Temporal uncertainty • Poisson process • Letting m = lt Then

  35. Probabilistic Seismic Hazard Analysis Temporal uncertainty 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? • l = 1/1000 = 0.001 • P = 1 - exp[-(0.001)(100)] = 0.0952

  36. Probabilistic Seismic Hazard Analysis Temporal uncertainty • 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 l,

  37. Probabilistic Seismic Hazard Analysis Temporal uncertainty Then, the annual rate of exceedance for an event with a 10% probability of exceedance in 50 yrs is The corresponding return period is TR = 1/l = 475 yrs. For 2% in 50 yrs, l = 0.000404/yr TR = 2475 yrs

  38. Probabilistic Seismic Hazard Analysis Summary of uncertainties Location Size Effects Timing fR(r) fM(m) P[Y > Y*| M=M*, R=R*] P = 1 - e-lt Source-site distance pdf Magnitude pdf Attenuation relationship including standard error Poisson model

  39. U U U P[A B1] + P[A B2] + … + P[A BN] B3 B2 B1 A B5 B4 Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations • P[A] = • P[A] = P[A|B1]P[B1] + P[A|B2]P[B2] + … + P[A|BN]P[BN] Total Probability Theorem

  40. Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations 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,

  41. Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations 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. where n = exp[a - bmo]

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

  43. Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations Dividing the range of possible magnitudes and distances into NM and NR increments, respectively This expression can be written, equivalently, as

  44. Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations What does it mean? All possible distances are considered - contribution of each is weighted by its probability of occurrence All possible magnitudes are considered - contribution of each is weighted by its probability of occurrence All sites are considered All possible effects are considered - each weighted by its conditional probability of occurrence

  45. rNR r1 m1 m2 m3 mNM Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations NM x NR possible combinations Each produces some probability of exceeding y* Must compute P[Y > y*|M=mj,R=rk] for all mj, rk

  46. P[Y > y*| M=m2, R=r1] ln Y M=m2 P[Y > y*| M=m2, R=r2] P[Y > y*| M=m2, R=r3] ln Y Y = y* r1 r2 log R r3 rN Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations Compute conditional probability for each element on grid Enter in matrix (spreadsheet cell)

  47. rNR P[Y > y*| M=m2, R=r3] P[Y > y*| M=m2, R=r2] P[Y > y*| M=m2, R=r1] r1 m1 m2 m3 mNM Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations • “Build” hazard by: • computing conditional probability for each element • multiplying conditional probability by P[mj], P[rk], ni • Repeat for each source - place values in same cells

  48. rNR P[Y > y*| M=m2, R=r3] P[Y > y*| M=m2, R=r2] P[Y > y*| M=m2, R=r1] r1 m1 m2 m3 mNM Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations • When complete (all cells filled for all sources), • Sum all l-values for that value of y* ly*

  49. rNR P[Y > y*| M=m2, R=r3] P[Y > y*| M=m2, R=r2] P[Y > y*| M=m2, R=r1] r1 m1 m2 m3 mNM Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations Choose new value of y* Repeat entire process Develop pairs of (y*, ly*) points Plot Seismic Hazard Curve log TR log ly* y*

  50. Probabilistic Seismic Hazard Analysis Combining uncertainties - probability computations 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. log lamax log TR log TR log ly* amax y*

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