SIGMA: What it is, why it ma-ers and what we can do with it

Transcription

1 Next Genera*on A-enua*on for CEUS (NGA East) Project SIGMA: What it is, why it ma-ers and what we can do with it Julian J Bommer Imperial College London

2 PGA PGV

3 M = 4.3 M = 6.0 M = 7.6

4 (J. Steidl)

5 Empirical Ground Mo*on Predic*on Equa*ons (GMPEs) Relate the logarithm of the chosen ground mo*on parameter, Y, to explanatory variables that characterize: EARTHQUAKE SOURCE Magnitude Style of Faul*ng SOURCE to SITE TRAVEL PATH Distance RECORDING SITE Surface geo materials

6 Empirical GMPEs log(y) = f(m, F, R, S) e.g., Akkar & Bommer (2010)

7 GMPE coefficients determined by regression analysis on recorded data

8 log(y obs ) log(y) δ, residual log(y pred ) M log(r)

9 δ = log(y obs ) log(y pred ) = log(y obs ) f(m, F, R, S) We introduce a third Greek le-er, ε, to represent the residuals normalized by the standard devia*on, ε= δ/σ ε Probability of Exceedance 0 50% (median) 1 16% 1 84% 2 2.3% 3 0.1%

10 The logarithmic residuals are generally found to conform to a normal (Gaussian) distribu*on with mean 0 and standard devia*on σ The distribu*on of the ground mo*on residuals can therefore be completely characterized by the logarithmic standard devia*on, σ

11 S I G M A CATTER N ROUND OTION TTENUATION log(pga) N(0,σ) M log(distance)

12 log(y) = f(m, F, R, S) + δ = f(m, F, R, S) + ε.σ log(pga) ε.σ N(0,σ) M log(distance)

13 Soil site recordings of September 2004 Parkfield earthquake ε = 2 ε = 1 ε = 0

14 Strong Mo*on Accelerogram

15 Value of σ varies with treatment of horizontal record components Beyer & Bommer (2006)

16 σ[ln(y)] = 2.3σ[log 10 (y)] ~300 records Year ~3000 records 0.15 to 0.35 log 10 [0.35 to 0.80 ln] Strasser et al. (2009)

17 Boore et al. (1997) equa*ons, median ±σ 84 percen*le PGA values generally about 80% > medians

18 Impact of σ on Seismic Hazard Analysis: DSHA (Kramer, 1996)

19 Impact of σ on Seismic Hazard Analysis: DSHA log(y) log(y 84 ) log(y 50 ) 84 th Percen*le σ log(y) Median log(r) Figure courtesy of F.O. Strasser

20 Impact of σ on Seismic Hazard Analysis: DSHA Rather than simply using the median or the 84 percen*le PGA, it would be more ra*onal to select ε on the basis of the associated probability of exceedance.. (Strasser et al., 2008) But that choice should be influenced by the recurrence rate of the scenario earthquake, in which case we re doing PSHA.

21 Impact of σ on Seismic Hazard Analysis: PSHA Calculate the mo*on at the site due to every feasible scenario (M R ε) and calculate the associated frequency

22 PSHA Fault Source Area Source 2 0.3g R 1A Area Source 1 Log(N) Log(PGA) Log(R) Annual Frequency M M

23 PSHA Fault Source Area Source 2 0.3g R 1A Log(PGA) +ε Area Source 1 Log(N) Log(R) Annual Frequency M M

24 PSHA Fault Source Area Source 2 0.3g R 1B Log(PGA) +2ε Area Source 1 Log(N) Log(R) Annual Frequency M M

25 Combina*ons of M R ε to Produce 0.3 g at the Site Scenario R (km) M ε f(m) f(ε) Frequency Σ Integrate over all possible magnitudes at all possible loca*ons over all sources, and consider all values of ε

26 Impact of σ on Seismic Hazard Analysis: PSHA PSHA Not PSHA (Abrahamson, 2000)

27 Value of σ exerts strong influence in PSHA Increasing σ Bommer and Abrahamson (2006)

28 Can the influence of σ be reduced by trunca*ng at ε max? Yes, but to result in an appreciable reduc*on of hazard, we need to truncate at 3 standard devia*ons (ε max = 3)

29 PSHA for Bay Bridge Figure courtesy of Norm Abrahamson

30 Strasser et al. (2008) EPRI study in 2006 concluded that there is no sta*s*cal basis, using current strong mo*on datasets, to truncate at less than 3 sigmas

31 Uncertainty in Ground Mo*on Predic*on Sigma is a measure of ALEATORY variability This means that it represents inherent RANDOMNESS (from alea, La*n for dice ) We could think of it, however, as apparent randomness since it the aleatory variability w.r.t. a model (GMPE)

32 Variability in Ground Mo*on Predic*on GMPEs are very simple (crude) models for very complex processes Therefore, a major contribu*on to σ is the absence of parameters that influence the ground mo*on but are not included in GMPEs e.g., SOURCE SIZE Magnitude included Stress drop, direc*vity, etc., etc., not e.g., SITE GEOLOGY V s30 used to characterize site effect Deeper geological structure o~en not

33 Sylmar County Hospital (Los Angeles) Nesher Site (Haifa)

34 SCH NES V s30 = 280 m/s V s30 = 284 m/s

35 Median amplifica*on func*ons from non linear site response analyses with 120 records Figure courtesy of Myrto Papaspiliou

36 Uncertainty in Ground Mo*on Predic*on Unless the data is abundant and well distributed with respect to the explanatory variables, there will be uncertainty regarding the posi*on of the median predic*on of ground mo*on. Bommer & Abrahamson (2007)

37 Uncertainty in Ground Mo*on Predic*on This is referred to as EPISTEMIC uncertainty because it reflects our lack of knowledge regarding earthquake source processes and wave propaga*on in the region under study (From epistêmê Greek for knowledge ) Celsus Library, Ephesus

38 Epistemic Uncertainty in GMPEs The epistemic uncertainty in the median ground mo*ons is usually incorporated into the hazard analysis through a logic tree, with branches carrying different models to which weights (reflec*ng the rela*ve confidence of the analyst in each model being the most appropriate for the region) are assigned Whereas the aleatory variability influences the shape of the hazard curve, the epistemic uncertainty results in several hazard curves

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40 MEAN

41 Median spectra for strike slip earthquakes recorded on rock sites at 10 km, from NGA models Abrahamson et al. (2008)

42 Epistemic Uncertainty in GMPEs In addi*on to the epistemic uncertainty in the median ground mo*ons predic*ons, there is also epistemic uncertainty associated with the value of sigma for each equa*on For example, there is s*ll uncertainty about whether sigma is dependent on earthquake magnitude or independent of magnitude

43 NGA models: Magnitude dependence of σ Abrahamson et al. (2008)

44 GMPEs with Heteroscedas*c Sigma Strasser et al. (2009)

45 Akkar & Bommer (2007) European GMPE Pure error analysis, following Douglas & Smit (2001), revealed apparently strong magnitudedependence of standard devia*on

46 Akkar & Bommer (2007) European GMPE

47 Akkar & Bommer (2010) Sigma Values

48 Epistemic Uncertainty in GMPEs Median, μ Sigma, σ GMPE 2 (w=0.2) GMPE 3 (w=0.4) Sigma 2 (w=0.4) σ µ σ σ

49 Can σ be reduced? In theory, since it represents inherent randomness, it is irreducible But σ is the apparent randomness in the observa*ons with respect to a par*cular model that a-empts to explain those observa*ons (i.e., it is the part that remains unexplained) Therefore, if we develop models that be-er explain the data, the apparent variability should decrease

50 Adding Explanatory Variables In addi*on to characterizing the earthquake source only by it size (magnitude), we can also include the influence of the style of faul*ng The impact on σ is modest, but nonetheless worthwhile Bommer et al. (2003)

51 Refine Explanatory Variables? Courtesy of Dave Boore

52 Courtesy of Dave Boore

53 Values of σ not reducing significantly over *me..

54 despite increase in the complexity of the equa*ons Data courtesy of J. Douglas

55 Empirical GMPE for Italy Bommer & Scherbaum (2005)

56 Norm Abrahamson 2009 EERI Dis*nguished Lecture As our knowledge of the genera*on and propaga*on of earthquake ground mo*on improves. Epistemic Uncertainty will be REDUCED Aleatory Variability will be REFINED The key to refining σ is its decomposi*on into different elements, and establishing the influences on each of these, and indeed whether they are all purely aleatory or if some of the components of σ are actually epistemic

57 σ T is the total variability τ is the inter-event (earthquake-toearthquake) variability Strasser et al. (2009) σ is the intra-event (record-to-record) variability

58 Akkar & Bommer (2010) European GMPE

59 The aleatory variability can be broken down into Modeling and Parametric components, σ m and σ p respec*vely: Bommer & Abrahamson (2007)

60 Errors in metadata (magnitudes, depths, distances, etc.) are propagated into the total variability. If these contribu*ons to the variability can be quan*fied, they can be subtracted from the appropriate variability component Abrahamson & Silva (2008) NGA model Strasser et al. (2009)

61 The Ergodic Assump*on In seismic hazard analysis, we are interested in the varia*ons in ground mo*on amplitudes at a par*cular site over *me (i.e., with repeated earthquakes). Since in general we do not have observa*ons over long periods at any site, we use records from many sites (and regions) to represent the variability of the ground mo*on The ergodic assump*on therefore is that temporal variability of ground mo*on can be represented by spa*al or even regional variability (i.e., trade space for *me)

62 Single Sta*on Sigma When we do have many recordings from a single sta*on, it is seen that the variability is smaller than the total sigma values calculated for standard GMPEs Using records from the LA basin, Atkinson (2006) found that single sta*on sigma values were, on average, 10% smaller than the σ calculated using all the sta*ons

63 Atkinson (2006)

64 Single Source Sta*on Sigma Using mul*ple recordings from a single site of earthquakes in only one par*cular source region (i.e., sampling a single travel path), much larger reduc*ons in σ have been found For LA basin case, Atkinson (2006) found that reduc*ons of up to 40% compared to σ values calculated using all the sta*ons and records from mul*ple sources

65 Midorikawa et al. (2008) 7,753 K Net and Kik Net records from 50 earthquakes (M w > 5.0) in 6 source zones

66 Lin et al. (2009) Using recordings from single sta*ons and single paths in Taiwan Figure courtesy of Norm Abrahamson

67 Region Total Single Site Single Path and site Chen & Tsai (2002) Taiwan Atkinson (2006) Southern CA Morikawa et al (2008) Japan Lin et al (2009) Taiwan Table courtesy of Norm Abrahamson But, this reduc*on in Aleatory Variability can only be invoked if the median mo*ons for the site/path known with confidence; otherwise, there is penalty to be paid in terms of increased Epistemic Uncertainty

68 σ σ μ trade off Strasser et al. (2009)

69 Es*ma*ng σ from numerical simula*ons The aleatory variability cannot be obtained simply by calcula*ng the residuals of the data with respect to the model The variability needs to be determined from the variability of the parameters in the simula*on models (taking account of their correla*ons to avoid over es*ma*on) and the misfit of model to observa*ons, reflec*ng the influence of parameters not included Useful to dis*nguish between MODELING uncertainty (due to the difference between the actual physical process genera*ng ground mo*ons and the simplified model represen*ng this process) and PARAMETRIC uncertainty (in the values of the parameters in the model for future earthquakes)

70 The modeling and parametric components of uncertainty can each be broken down into ALEATORY and EPISTEMIC components, leading to four components of the total uncertainty in ground mo*on predic*ons: Toro et al. (1997)

71 e.g., variability in Δσ for CEUS earthquakes e.g., uncertainty in median Δσ and its variability for CEUS earthquakes

72 ALEATORY: Parametric uncertainty in stress drop, focal depth, κ and Q, and from aleatory modeling uncertainty. EPISTEMIC: Epistemic parametric uncertainty in stress drop, and from epistemic modeling uncertainty. Toro et al. (1997)

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75 Concluding Remarks Sigma is the aleatory variability in ground mo*on predic*ons It is part and parcel of the GMPE; the median predic*on alone is not a full representa*on Sigma cannot be wished away, must always be taken into account and it has a major impact on the results PSHA Trunca*ng sigma is not a feasible op*on to reduce its influence in PSHA at the current *me The most promising prospects for reducing sigma, and its impact, is to break it down into components, iden*fy those that are actually epistemic, take them out and deal with them separately through data collec*on/modeling or logictree branches

76 Decomposing σ into Aleatory & Epistemic Components PEGASOS Refinement Project mee*ng, London, January 2010