Sigma: Issues, Insights & Challenges

1. NGA-East Sigma Workshop – PEER, UC Berkeley, 30 September 2009 Sigma: Issues, Insights & Challenges Fleur O. Strasser Seismology Unit Council for Geoscience South Africa fstrasser@geoscience.org.za

2. Seismological Research Letters 79(2), 40-56 Jan/Feb 2009

3. Overview • Review of state-of-the-knowledge regarding sigma (SRL paper) • Response to comments on SRL paper • Sigma in recent GMPEs for SCRs

4. PGA Accelerogram Acceleration Time Soil site Rock site N(0,σ) Propagation path • Explanatory variables: • Magnitude (M) • Distance (R) • Site conditions • etc.. Seismic source Aleatory ground-motion variability (σ) Predictive equations obtained by regression: Log(Y) Scatter (σ) = Aleatory variability M Log(R) + εσ log Y = f(M, R, site classification,..) Ground-motion parameter Median prediction based on explanatoryvariables Residual σrepresents apparent randomness of the ground-motion w.r.t. comparatively simple model – “intrinsic” randomness is expected to be less

5. Bommer and Abrahamson (2006) Increasing s Why is σ important? DSHA PSHA Ground motion for a given (M,R,ε) scenario Integration over all possible (M,R,ε) scenarios σ controls the shape of the hazard curves σ required to select the design level ground motion Log(Y) 84th Percentile Log(Y84) Log(Y50) σLog(Y) Median M Log(R)

6. Typical values of σ No significant reduction in σ despite increase in number of recordings and advances in ground-motion modelling 0.15 to 0.35 log10 0.35 to 0.80 ln After Douglas (2003)

7. Factors influencing σ log Y = f(M, R, site classification,..) + εσ PREDICTED VARIABLE • Selection of GM parameter • Horizontal component definition (5-10% adjustment) • Quality of parameter estimates (usable frequency range + processing) • Data Selection

8. Factors influencing σ log Y = f(M, R, site classification,..) + εσ PREDICTED VARIABLE FUNCTIONAL FORM • Selection of GM parameter • Horizontal component definition • Quality of parameter estimates (usable frequency range + processing) • Data Selection • Physical processes included • Level of complexity of the model (e.g., soil non-linearity) EXPLANATORYVARIABLES • Parameter definition (e.g., point-source vs. finite-fault distance metrics) • Quality of the parameter estimates (“METADATA ERRORS”)

9. Metadata Errors Metadata errors contribute to variability through error propagation  Importance of selecting recordings with good-quality metadata Abrahamson and Silva (2008) NGA Model

10. Issue 1: Unbalanced Datasets • In a given dataset, the number of recordings from individual events is very variable • Limited number of multiple events recorded at the same station.  Affects estimation of individual variability components 1-way classification (block-diagonal covariance matrix) 2-way classification (covariance matrix no longer block-diagonal) Station-specific variability Record-specific variability

11. Issue 2: Is σ homoscedastic or heteroscedastic? • Magnitude • Decrease of σ with increasing magnitude found by several authors • Suggested reasons: • decrease in stress drop variability • soil non-linearity • differences in GM scaling not covered by equation • Distance • No significant dependence found except in the very near-source region • Distance-dependence may result from dependence on other factors controlling ground-motion amplitude • Other variables • In AS08 NGA model, dependence on VS,30 and depth through propagation of metadata errors

12. Issue 3: σ on soil vs. σ on rock • σ on soil generally considered to be smaller than σ on rock due to soil non-linearity • However, exceptions in the case where other physical processes influencing GM variability affect selectively soil residuals (e.g., basin effects) • Explicit consideration of soil non-linearity effects in the equation results in a reduction of σ at short distances • Rate of distance-dependence depends on modelling of non-linearity effects Abrahamson and Silva (2008): based on 1-D site response model Chiou and Youngs (2008): direct regression on residuals

13. Issue 4: Trade-offs in the estimation of σ and μ • σ is an indicator of the level of confidence associated with the estimate of the median • Affected by composition of the dataset • Combining datasets with different medians (e.g., to increase the number of data points considered) may result in a broadening of the overall distribution  INCREASING NUMBER OF RECORDINGS IN DATASET MAY INCREASE σIN SOME CASES

14. Issue 5: Is σ region-dependent? • Sometimes considered better to limit recordings included in dataset to those obtained in a limited geographical region. • Douglas (2007) found that σ values for response ordinate equations derived using data from small geographical areas are generally higher than σ values for equations derived combining data from several areas • Example: Liu and Tsai (2005) • Taiwan as a whole: σ = 0.687 • CHY region only : σ = 0.637 • NTO region only : σ = 0.685 • IWA region only: σ = 0.703 • Higher values for small regions likely to be related to data limitations Liu and Tsai (2005)

15. Challenge: Can we reduce σ? • ...by adding more data? ~ 300 data points ~ 3000 data points

16. Challenge: Can we reduce σ? • ...by using more sophisticated models? • Comparison of NGA vs. previous generation of models shows: • some reduction at longer periods (soil non-linearity models) • overall, similar (or even higher values) of σ  INTRODUCING ADDITIONAL PARAMETERS CARRIES A PENALTY IN TERMS OF PARAMETRIC UNCERTAINTY

17. Challenge: Can we tailor σ to a specific project? • RATIONALE: Not all the physical factors controlling GM variability are necessarily relevant to a specific project. Eliminating irrelevant factors could reduce the value of σ to be used for that specific project. • EXAMPLES: • Atkinson (2006) LA Basin • station-specific σ: ~10% reduction • station and source-specific σ: ~40% reduction • Morikawa et al. (2008) Japan • ~50% reduction using source-site specific correction • Lin et al. (2009) Taiwan • 30% reduction using path correction factor • BUT: PENALTY TO PAY IN TERMS OF EPISTEMIC UNCERTAINTY ON THE MEDIAN  seismic hazard values are not necessarily reduced !

18. Conclusions – Sigma Paper • Estimating σ is an inescapable reality of the ground-motion prediction process, and an integral component of SHA. • Despite numerous attempts at reducing it, the value of σ has remained fairly stable over the past 40 years. • Since not all contributors to σ are related to intrinsic randomness of the ground-motion, there is some scope to achieve a reduction through the careful selection of data and the use of more realistic physical models. However, the reductions thus achieved have been small. • The most promising approaches to reduce σ consist in getting better constraints on the individual components of variability and “tailoring” the estimation to the specific project of interest. • However this carries a penalty in terms of epistemic uncertainty on the median ground motion and thus does not necessarily lead to reduced seismic hazard values.

19. Seismological Research Letters 80(3), 491-493 Seismological Research Letters 80(3), 494-498

20. Wang Comment – Sigma Paper δ=δ(m,r) σ= σ(m,r) Wang: Hazard integral above valid if and only if M,R and δ are independent random variables – not the case since δ= δ(m,r) hence PSHA mathematically invalid • Hazard integral considers ε, not δ • M,R not independent variables in general • However ε (normalised residual) is independent of M and R (explanatory variables) as a consequence of the regression process – whether σ is homoscedastic or not (e.g., Bazzurro & Cornell, 1999)

21. Klügel Comment – Sigma Paper Strasser et al. (2009): Klügel et al. (2006) find metadata errors (magnitude) have a large influence on σ – but they use a standard deviation of 0.4 magnitude units based on bin widths and uncertainty from historical catalogue data. BUT: Datasets used for GMPE derivation would not consider data with such uncertain metadata Recently, Klügel et al. (2006) have suggested that measurement errors in magnitude have a large impact on the standard deviation. This conclusion was based on an assumption that there is a very large uncertainty of the magnitude (standard deviation of 0.4 magnitude units). Klügel et al. (2006) justified their use of this large standard deviation of the magnitude estimate by the approach they proposed for scenario-based risk calculations. In this approach, Klügel et al. (2006) grouped the earthquakes into broad magnitude bins to reduce the number of scenarios to be considered, and based the magnitude uncertainty of 0.4 units on the width of these broad magnitude bins. However, the measurement error of magnitudes in ground-motion databases is a seismological issue based on regional and teleseismic recordings of earthquakes; it is not dependent on how earthquake magnitudes are grouped in an engineering application. Therefore, the results given in Klügel et al. (2006) showing a large impact on the estimated standard deviation due to measurement errors in magnitude are not credible. • Assessment (and removal) of measurement (metadata) errors is important • Issue of uncertainty associated with scenario-based calculations – how do we estimate this?

22. CEUS GMPEs Sigmas Significantly different models!!

23. Raghu Kanth & Iyengar (2007)Peninsular India Bedrock median GM and sigma modified using randomised soil profiles corresponding to specific NEHRP site classes

24. UK GMPE (Rietbrock, Strasser & Edwards, in preparation) • GMPE from stochastic simulations calibrated using UK weak-motion data • Parameters derived using Q-tomography, summarised in Edwards et al. (2008) • Consider both “constant” and magnitude-dependent stress parameter models • Preserve {fc, Ω, t*} covariance structure => trade-offs

25. ALEATORY PARAMETRIC

26. Thank you.