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A Pseudo-Dynamic Rupture Model Generator for Earthquakes on Geometrically Complex Faults Daniel Trugman, July 2013

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2D Rough-Fault Dynamic Simulations Homogenous background stress + complex fault geometry heterogeneity in tractions Eliminates important source of uncertainty: fault geometry is a direct observable

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Rough Fault (not to scale)

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Pseudo-Dynamic Source Model Rough-fault simulations: high-frequency motions consistent with field observations But: too computationally intensive to incorporate into probabilistic hazard analysis Idea: use insight from rough-fault simulations to build a pseudo-dynamic source model – Source parameters consistent with dynamic models – Retain computational efficiency of kinematic models

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Method: Building a Pseudo-Dynamic Model Step 1: Study dynamic source parameters Step 2: Represent pseudo-dynamic source parameters as spatial random fields that are consistent with dynamic simulations Step 3: Compare source models and simulated ground motion for different fault profiles

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Step 1: Analyze Dynamic Source Parameters Δu, v rup, V peak – Mean, standard deviations – Autocorrelation: spatial coherence – Dependence on fault geometry Shape of source-time function, V(t) Restrict attention to: – subshear ruptures (background stress just high enough for self-sustaining ruptures) – region away from the hypocenter (nucleation zone)

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Source parameters are strongly anti-correlated with fault slope m(x):

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Source-time function of the form:

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Step 2: Represent pseudo-dynamic source parameters as spatial random fields: Assume Gaussian marginals – Use mean, standard deviations from dynamic simulations – Key step: anticorrelate with fault slope Assume exponential ACF: – Correlation length β from dynamic sims – V peak, Δu more spatially coherent than v rup – Power spectrum ~ k -2

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1. Generate four vectors of Gaussian noise (vector = discretization of fault profile) 2. Create fault profile h(x): filter first vector in Fourier domain with appropriate PSD (fractal) 3. Correlate other 3 vectors (the source parameters) with fault slope m(x) = dh/dx 4. Filter these 3 vectors to obtain desired PSD (e.g. exponential), scale to obtain desired 5. Combine source parameters and source-time function to form full slip- velocity function V(x,t) Basic rupture generating procedure:

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Step 3: Model Comparison Start with a direct comparison on a single (random) fractally-rough fault profile – Source parameters and seismic wave excitation – Also compare with flat-fault projection of pseudo- dynamic source parameters Generalize to ensemble comparison – 30 different (random) fractally-rough fault profiles

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final slip, Δu correlation coefficient: 0.80 rupture velocity, v rup correlation coefficient: 0.64 peak slip velocity, V peak correlation coefficient: 0.78 Source Parameters

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fault-parallel velocity (v x )fault-normal velocity (v y ) Seismograms

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dynamic simulationpseudo-dynamic simulation Seismic Wavefield (fault-normal velocity)

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rough fault pseudo-dynamic simulation flat fault pseudo-dynamic simulation Seismic Wavefield (fault-normal velocity)

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Ensemble Marginal Distributions: Δu

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Ensemble Marginal Distributions: v rup

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Fourier Amplitude Spectra (fault-normal acceleration)

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Peak Ground Acceleration

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Discussion: generalization to 3D 2D autocorrelation structure – i.e β x and β z Which slope to use? – Trace of the fault plane in the slip direction? Component of rupture velocity in z direction? – No correlation with z-direction slope (given stress field)? Need to taper source parameter distributions at source boundaries? Thrust faults? – Which is the relevant slope? – Is this different for rupture velocity than for slip?

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Extra Slides:

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Conclusions Fault geometry strongly influences rupture process and hence, the earthquake source parameters. Our pseudo-dynamic model produces comparable ground motion to that seen in dynamic models, even at high frequencies. Similar models could be implemented in programs like CyberShake to improve our understanding of seismic hazard.

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Figure References Dunham, E.M., Belanger, D., Cong, L., and J.E. Kozdon (2011). Earthquake ruptures with strongly rate-weakening friction and off- fault plasticity, Part 2: Nonplanar faults, BSSA, 101, no. 5, 2308- 2322, doi: 10.1785/0120100076. Graves, R. et al. (2011). CyberShake: A physics-based seismic hazard model for southern California, Pure Appl. Geophys., 168, no. 3-4, 367-381, doi: 10.1007/s00024-010-0161-6. Sagy, A.,Brodsky, E. E., and G. J. Axen (2007). Evolution of fault- surface roughness with slip, Geology, 35, 283-286, doi: 10.1130/G23235A.1 Shi, Z., and S. M. Day (2013). Rupture dynamics and ground motion from 3-D rough-fault simulations, J. Geophys. Res. (in press). Song, S. G. and L. A. Dalguer (2013). Importance of 1-point statistics in earthquake source modelling for ground motion simulation, Geophys., J. Int., 192, no.3, 1255-1270, doi: 10.1093/gji/ggs089

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Ensemble Marginal Distributions: V peak

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Peak Ground Velocity

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Fourier Amplitude Spectra

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Basic Procedure: 1.Generate fault profile h(x) (filter Gaussian noise in Fourier domain to obtain correct PSD) 2.Correlate source parameter vectors with m(x) 3.Filter correlated vectors to achieve desired PSD 4.Rescale and shift: correct mean and std. dev. 5.Aggregate source parameters V(x,t)

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Complex Fault Geometry Most dynamic rupture simulations assume planar faults, model stress field as random field But faults are fractally rough: deviate from planarity at all length scales: Sagy et al. Geology 2007; 35: 283-286 Dixie Valley Fault, Nevada

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