Ground-Motions for Regions Lacking Data from Earthquakes in M-D Region of Engineering Interest

1. Ground-Motions for Regions Lacking Data from Earthquakes in M-D Region of Engineering Interest • Most predictions based on the stochastic method, using data from smaller earthquakes to constrain such things as path and site effects

2. Simulation of ground motions • representation theorem • source • dynamic • kinematic • path & site • wave propagation • represent with simple functions

3. Representation Theorem Computed ground displacement

4. Representation Theorem Model for the slip on the fault

5. Representation Theorem Green’s function

6. Types of simulations • Deterministic • Deterministic description of source • Wave propagation in layered media • Used for lower frequency motions • Stochastic • Random source properties • Capture wave propagation by simple functional forms • Can use deterministic calculations for some parts • Primarily for higher frequencies (of most engineering concern)

7. Types of simulations • Hybrid • Deterministic at low frequencies, stochastic at high frequencies • Combine empirical ground-motion prediction equations with stochastic simulations to account for differences in source and path properties (Campbell, ENA) • Empirical Green’s function

8. Stochastic simulations • Point source • With appropriate choice of source scaling, duration, geometrical spreading, and distance can capture some effects of finite source • Finite source • Many models, no consensus on the best (blind prediction experiments show large variability) • Often incorporate point source stochastic model

9. The stochastic method • Overview of the stochastic method • Time-series simulations • Random-vibration simulations • Target amplitude spectrum • Source: M0, f0, Ds, source duration • Path: Q(f), G(R), path duration • Site: k, generic amplification • Some practical points

10. Stochastic modelling of ground-motion • Deterministic modelling of high-frequency modelling not possible except in the far-field (lack of Earth detail and computational limitations) • Hanks & McGuire (1981): Incoherent character of HF ground motion can be captured by representing it as band-limited, finite-duration Gaussian white noise (captures the essence of the physical process) • Several methods use stochastic representation • D. Boore (1983 -) • Papageorgiou & Aki (1983) • Zeng et al. (1994) • NB: Stochastic method = stochastic model

11. SMSIM Programs (1983-) • SMSIM = Stochastic Model SIMulation • Series of FORTRAN programs based on : • Hanks & McGuire (1981) • Brune's point-source model (1970) • Idea: combine deterministic target amplitude obtained from simple seismological model and random phase to obtain realistic high frequency motion. Try to capture the essence of the physics using simple functional forms for the seismological model

12. Basis of stochastic method • Radiated energy described by the spectra in the top graph is assumed to be distributed randomly over a duration given by the addition of the source duration and a distant-dependent duration that captures the effect of wave propagation and scattering of energy • These are the results of actual simulations; the only thing that changed in the input to the computer program was the moment magnitude (5 and 7)

13. SMSIM Programs (1983-) - continued • Ground motion and response parameters can then be obtained via two separate approaches: • Time-series simulation: • Superimpose a random phase spectrum on a deterministic amplitude spectrum and compute synthetic record • All measures of ground motion can be obtained • Random vibration simulation: • Probability distribution of peaks is used to obtain peak parameters directly from the target spectrum • Very fast • Can be used in cases when very long time series, requiring very large Fourier transforms, are expected (large distances, large magnitudes) • Elastic response spectra, PGA, PGV, PGD, equivalent linear (SHAKE-like) soil response can be obtained

14. Time-domain simulation

15. Aim: Signal with random phase characteristics Probability distribution for amplitude Gaussian (usual choice) Uniform Array size from Target duration Time step (explicit input parameter) Step 1: Generation of random white noise

16. Aim: produce time-series that look realistic Step 2: Windowing the noise • Windowing function • Boxcar • Cosine-tapered boxcar • Saragoni & Hart (exponential)

17. FFT algorithm Step 3: Transformation to frequency-domain

18. Divide by rms integral Aim of random noise generation = simulate random PHASE only Step 4: Normalisation of noise spectrum Normalisation required to keep energy content dictated by deterministic amplitude spectrum

19. = Y(M0,R,f) E(M0,f) P(R,f) S(f) I(f) Instrument or ground motion Earthquakesource Propagation path Site response TARGET FOURIER AMPLITUDE SPECTRUM Step 5: Multiply random noise spectrum by deterministic target amplitude spectrum Normalised amplitude spectrum of noise with random phase characteristics

20. Numerical IFFT yields acceleration time series Manipulation as with empirical record 1 run = 1 realisation of random process Single time-history not necessarily realistic Values calculated =average over N simulations (50 < N< 200) Step 6: Transformation back to time-domain

21. Steps in simulating time series • Generate Gaussian or uniformly distributed random white noise • Apply a shaping window in the time domain • Compute Fourier transform of the windowed time series • Normalize so that the average squared amplitude is unity • Multiply by the spectral amplitude and shape of the ground motion • Transform back to the time domain

22. Effect of shaping window on response spectra

23. Acceleration, velocity, oscillator response for two very different magnitudes, changing only the magnitude in the input file

24. Warning: the spectrum of any one simulation may not closely match the specified spectrum. Only the average of many simulations is guaranteed to match the specified spectrum

25. Random Vibration Simulation

26. Random Vibration Simulations - General • Aim: Improve efficiency by using Random Vibration Theory to model random phase • Principle: • no time-series generation • peak measure of motion obtained directly from deterministic Fourier amplitude spectrum through rms estimate

27. is Fourier Transform of • Predicts peak time domain motion from frequency domain • Makes use of Parseval’s Theorem

28. is acceleration time series is root-mean-square acceleration is ground motion duration is Fourier amplitude spectrum of ground motion • arms is easy to obtain from amplitude spectrum: • But need extreme value statistics to relate rms acceleration to peak time-domain acceleration

29. m2, m0 are spectral moments

30. Special consideration needs to be given to choosing the proper duration T to be used in random vibration theory for computing the response spectra for small magnitudes and long oscillator periods. In this case the oscillator response is short duration, with little ringing as in the response for a larger earthquake. Several modifications to rvt have been published to deal with this.

31. Comparison of time domain and random vibration calculations, using two methods for dealing with nonstationary oscillator response. • For M = 4, R = 10 km

32. Comparison of time domain and random vibration calculations, using two methods for dealing with nonstationary oscillator response. • For M = 7, R = 10 km

33. Target Amplitude Spectrum

34. Basic Assumption • Ground motion can be modeled as finite-duration bandlimited Gaussian noise, whose underlying spectrum can be specified (based on a seismological model) THE KEY TO THE SUCCESS OF THE MODEL LIES IN BEING ABLE TO DEFINE FOURIER ACCELERATION SPECTRUM AS F(M, DIST)

35. Parameters required to specify Fourier accn as f(M,dist) • Model of earthquake source spectrum • Attenuation of spectrum with distance • Duration of motion [=f(M, d)] • Crustal constants (density, velocity) • Near-surface attenuation (fmax or kappa)

36. Stochastic method • To the extent possible the spectrum is given by seismological models • Complex physics is encapsulated into simple functional forms • Empirical findings can be easily incorporated

37. Earthquakesource Propagationpath Site response Instrument or ground motion Target amplitude spectrum Deterministic function of source, path and site characteristics represented by separate multiplicative filters

38. Source Function

39. Source function E(M0, f) Source DISPLACEMENT Spectrum Scaling of amplitude spectrum with earthquake size • Scaling constant • near-source crustal properties • assumptions about wave-type considered (e.g. SH) Seismic moment Measure of earthquake size

40. Scaling constant C • βs = near-source shear-wave velocity • ρs = near-source crustal density • V = partition factor • (Rθφ) = average radiation pattern • F = free surface factor • R0 = reference distance (1 km).

41. Brune source model • Brune’s point-source model • Good description of small, simple ruptures • "surprisingly good approximation for many large events".(Atkinson & Beresnev 1997) • Single-corner frequency model • High-frequency amplitude of acceleration scales as:

42. Semi-empirical two-corner-frequency models • Aim: incorporate finite-source effects by refining the source scaling • Example: AB95 & AS00 models • Keep Brune's HF amplitude scaling • fa, fb and e determined empirically (visual inspection & best-fit)

43. The spectra can be more complex in shape and dependence on source size. These are some of the spectra proposed and used for simulating ground motions in eastern North America. The stochastic method does not care which spectral model is used. Providing the best model parameters is essential for reliable simulation results (garbage in, garbage out).

44. Scaling of the source spectrum • Based on Aki’s (1967) ω²-model • Single corner frequency • For acceleration: • LF: ω² increase proportional to M0 • HF: constant amplitude, depends on M0,  as shown • Self-similar scaling • The key is to describe how the corner frequencies vary with M. Even for more complex sources, often try to relate the high- frequency spectral level to a single stress parameter

45. Stress parameter: introduction • “Seismologists argue over everything about this parameter, including its name, meaning, measurement, and scaling with magnitude”. (Atkinson & Beresnev 1997 – "Don't call it stress drop") • Different notions covered by single term/notation • Non-unique determination even for Brune model (arbitrary assumptions required to represent finite fault by a point-source model) • Physical meaning for 2-corner-frequency models not straightforward

46. u = average slip r = characterisic fault dimension Stress parameter: definitions • "Stress drop" should be reserved for static measure of slip relative to fault dimensions • "Brune stress drop" = change in tectonic (static) stress due to the event • "SMSIM stress parameter" = “parameter controlling strength of high-frequency radiation” (Boore 1983)

47. Stress parameter:treatment in SMSIM • Allowed to vary linearly with magnitude • Self-similar scaling breaks down at higher magnitudes • Possibly trade-off with upper-crust attenuation (k) • Most applications use constant Ds • Required for single-corner Brune model • Trial & error approach: all other parameters fixed, multiple runs to find best estimate • Characteristic of regional geology (shattered vs. competent rock) • Brune stress drop used as guideline for trial & error

48. Brune stress drop Hanks & Kanamori (1975) 10 to 100 bar 30 bar for inter-plate events 100 bar for intra-plate events overall average of 60 bar Boore & Atkinson (1987) 1 to 200 bar for moderate to large earthquakes literature review compiling the opinion of several authors Ds is “apparently independent of source strength over 12 orders of magnitude in seismic moment”. SMSIM stress parameter California: 50 - 100 bar ENA: 150 – 200 bar (greater uncertainty) High values due to finite-fault & other effects (asperity) not included in point-source model Stress parameter - Values

49. Source duration • Required to define array size (both TD & RV) • Determined from source scaling model via: • For single-corner model, fa= fb = f0 • Weights wa and wb should add up to the distance-indepent coefficient in the expression giving total duration

50. Path Function