5. Probabilistic Seismic Hazard Analysis

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1 Probabilistic Seismic Hazard Analysis (PSHA) proposed by C.A. Cornell (1968) used to determine the design earthquake for all locations in USA. PSHA gives a relative quantification i of the design earthquake, for a given site, in terms of seismic parameters (peak ground acceleration, peak ground velocity, peak spectral acceleration at particular periods etc.). PSHA takes into account uncertainties associated with occurrence and effects of important earthquakes. PSHA cannot predict the date, magnitude, location and effects of future earthquakes in active seismic zones. PSHA provides a basis for the seismic design of structures for a given location. PSHA used to construct US National Seismic Hazard Maps 27 Description The probabilistic seismic hazard analysis method, originally proposed by C.A. Cornell (1968), attempts to answer the following question: What is the probability per annum of exceeding a given ground motion (peak ground acceleration, velocity, etc.) at a given site? 28 1

2 Description Usually, the application of the probability seismic hazard analysis requires six separate steps: Step 1. Definition of the seismogenic sources For a given site, geographic zones representing seismotectonic sources are drawn. For each zone, it is assumed that the probability of earthquake occurrence is the same for the entire surface area (seismogenic source). Step 2. Definition of a seismicity model For each source, a magnitude-recurrence relation of the type log N = A bm is defined. Historical seismicity is used to establish the parameters of this relation. A cut-off magnitude, M max, is also defined for each source. Step 3. Definition of an attenuation model An attenuation relation is determined between each source and a given site. Often the same attenuation relation is used for the entire region. Step 4. Calculation of the recurrence rate of a given ground motion level The attenuation relations are combined with the magnitude-recurrence relations to find the annual number of earthquakes causing a chosen ground motion level at any ygiven site. Step 5. Calculation of the annual probability of exceeding a chosen ground motion level A theoretical probability distribution is used to calculate the annual probability of exceeding a chosen ground motion level at any given site. The calculation is based on the recurrence rate calculated in Step 4. Step 6. Plotting of contour lines having a constant annual probability of exceedence Steps 1 to 5 are repeated for each given site. Contour lines are then plotted for the chosen ground motion parameter. These contour lines are based on the same annual probability of exceedence for the entire surface area. 29 To illustrate the probabilistic seismic hazard analysis procedure, consider a site at the centre of a very large seismogenic source 30 2

3 Let us assume also that the magnitude-recurrence relation per unit area of the source is known. Equation 3.6 is inverted : where log N = A - b M (3.6) N = N o exp - b M (3.7) A N o= exp (3.8) The symbol «exp» is considered as the general form of the inverse of a logarithm. exp = e if log = log = ln exp =10 if log = log e 10 (3.9) Note that N is the number of earthquakes with magnitude greater or equal to M, per annum and per unit area. 31 Suppose that the parameter of interest at the site, is the peak ground horizontal acceleration, amax. A general attenuation relation, for the region, is given by equation 3.5. log a max = b1+b2 M - b3 log d + b4 (3.10) This equation is inverted to isolate amax. - exp M max = exp b1 b2 d +b b3 (3.11) a 4 where d represents the hypocentral distance. 2 2 d = R + h (3.12) where R = epicentral distance h = focal depth considered constant for the entire source. Note: deterministic attenuation relation 32 3

4 To simplify, the parameters of equation 3.11 are rewritten. Equation 3.11 is rewritten as follows: exp b1 = c b4 = 0 Now the equation is inverted in terms of M b2 M 1 (3.13) amax = c1 exp R + h 2 (3.14) M = b3 a b3 log 2 2 R + h 2 max c1 (3.15) b

5 Now, consider an infinitesimal annular element of source area da (fig. 3.4). If a unit source area generates N earthquakes per year of magnitude greater or equal to M, then an infinitesimal source area da generates dn earthquakes per annum of magnitude greater or equal to M. Equation 3.7 is written for dn earthquakes: dn = N o exp - b M da (3.16) 35 The surface da can be written as a function of the radius, R, from the site. da = 2 R dr (3.17) Equation 3.17 is substituted in equation b M R dr dn = 2 N o exp (3.18) 36 5

6 Substituting equation 3.15 into equation 3.16 leads to the number of annual earthquakes from surface da causing, at the site, a peak ground acceleration greater or equal to amax (the recurrence rate). a b b log R + h 2 max c1 dn = 2 N o exp b 2 R dr (3.19) This equation can be simplified as : a 2 2 dn = 2 N R + h b 3 b2 o 2 max R dr (3.20) c1 -b

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9 Let s assume that the magnitude-recurrence relation per unit area of the seismogenic source is defined by: A = and b = 1.85 log N = A - b M Consider 2 different attenuation relations Constant Constant Constant exp(a) exp(b1) 2No(amax/c1)^( b/b2) (b3*b/b2) 2 c1 exp(b2*mmax) h^( b3) 43 1/N Attenuation #2 Attenuation #1 44 9

10 Probability of Exceedence 45 Probability of Exceedence 46 10

11 General Application 47 General Application e.g. a max 48 11

12 General Application e.g. a max Note: Uncertainty in attenuation relationships included by approximating, for a given sector i, the number of earthquakes per annum N i of magnitude between m and m + dm and capable of producing a peak acceleration greater or equal than a max N N ( m) N ( m dm) P PHA a i i i max 49 (deterministic) 50 12

13 51 Model Uncertainty PSHA provide a systematic framework for the treatment of uncertainty in the values of the parameters of a particular seismic hazard model however, best choices for elements of hazard model are themselves uncertain use of logic trees provides a convenient framework for the explicit treatment of model uncertainty 52 13

14 Model Uncertainty each alternative model assigned a weighting factor likelihood of that model being correct tree consists of a series of nodes points at which models are specified and branches that represent the different models under consideration Sum of probabilities of all branches connected to a node must be 1.0 hazard analysis carried out for each combination of models associated with a terminal branch result of each analysis weighted by relative likelihood of its combination of branches final result is the sum of the weighted individual results 53 Hazard Curve Deaggregation combinations of magnitude and distance that contribute most to hazard curve useful for selecting earthquake histories for response N analysis m M m m; r R r r C( m, r) N total 54 14

15 6. USGS Seismic Hazard Maps United States Geological Survey (USGS) National Hazard Maps show distribution of earthquake shaking levels that have a certain probability of occurring in the United States. Maps created to provide most accurate and detailed information possible to assist engineers in designing buildings, bridges, highways, and utilities that will withstand shaking from earthquakes in the United States. Maps are used to create and update building codes Used by more than 20,000 cities, counties, and local governments to help establish construction requirements necessary to preserve public safety USGS Seismic Hazard Maps Applications of the Hazard Maps: 1. Building Codes (NEHRP, IBC, ASCE 7) 2. Highway bridge design nationwide (AASHTO) 3. Insurance rates 4. Business and land-use planning 5. Estimations of stability and landslide potentials of hillsides 6. Construction standards for waste-disposal facilities (EPA) 7. Retrofit priorities 8. Allocation planning of assistance funds for education and preparedness (FEMA) 9. Concerned general public

16 6. USGS Seismic Hazard Maps 2002 maps are for: Peak Ground Acceleration (PGA) sec and d sec spectral acceleration (for 5% damping) 10% and 2% probabilities of exceedance (PE) in 50 years Web site: s/design/index.php USGS Seismic Hazard Maps 58 16

17 7. References Bozorgnia, Y. and Bertero, V. V. (Eds), (2004). Earthquake Engineering: From Engineering Seismology to Performance-Based Engineering, CRC Press, Boca Raton, FL. Cornell, C.A. (1968). Engineering Seismic Risk Analysis, Bulletin of the Seismological Society of America, 58(5); EERI Committee on Seismic Risk. (1984). Glossary of Terms for Probabilistic Seismic Risk and Hazard Analysis, Earthquake Spectra, 1(1); Gutemberg, B. and Richter, C.G. (1965). Seismicity of the Earth, New York, Hafner. Kaila, K.L. ad Narain, H. (1971). A New Approach for the Preparation of Quantitative Seismicity Maps, Bulletin of the Seismological Society of America, 61(5); Kramer, S. (1996). Geotechnical Earthquake Engineering, Prentice Hall, NJ. McGuire, R.K. (1976). Fortran Computer Program for Seismic Risk Analysis, Open Report 76-67, US Geological Survey, Washington, DC. USGS National Seismic Hazard Maps, Data, and Documentation, US Geological Survey, Available online at 59 Questions/Discussions 60 17