Antonella Cirella 1, Paul Spudich 2 1 INGV, Rome, Italy 2 USGS, Menlo Park, CA Aleatory and Epistemic Uncertainties in Interpolated Ground Motions Example.

Antonella Cirella1, Paul Spudich2

1INGV, Rome, Italy2USGS, Menlo Park, CA

Aleatory and Epistemic Uncertainties in Interpolated Ground Motions

Example from the Kashiwazaki-Kariwa Nuclear Power Plant Recordings of the July 16, 2007, Niigata-ken Chuetsu-oki,

Japan, Earthquake

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Using an earthquake rupture model to ‘predict’ ground motions at unobserved

locationsWhen earthquakes occur, we could have a few recordings of ground motion, but not a sites where something interesting and /or bad occurred. (e.g. 2007 Niigata-ken Chuetsu-oki)

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locationsPost-Earthquake forensic studies and ‘Shake-Map’ generation rely upon ground-motion interpolation (process to infer, from a set of ground motions, observed at sparse observation locations, the ground motion at a target site where no observation were made during the earthquake)

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locations

It is now possible to invert observed ground motions to obtain the spatio-temporal behavior of rupture on

the causative fault

Forward predict ground motions at unobserved locations

But the inversion is inherently non-unique How large is the scatter in the interpolated

ground motions?

Is the scatter of the predicted motions large enough to include the true motions at the target sites?

Is the scatter of the predicted motions at the unobserved site larger than the scatter of the predicted motions at sites having data used in the inversion?

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A problem in linear inference, except…

• Theoretically, the process of ground-motion interpolation is an example of linear inference.

• For a linear problem lacking bounds on the model space, the interpolated ground motion can be completely unbounded.

• When bounds like positivity and known moment are added as constraints on the model, the interpolated motions may then become bounded, and could be explored using algorithm BVLS of Stark and Parker (1995), which can compute bounds on linear functionals of an inverted model.

• However, the response spectrum is a nonlinear function of the ground acceleration time series, and consequently we have chosen to investigate the variability of interpolated response spectra by using a statistical approach building on a non-linear inversion method (Piatanesi et al, 2007; Cirella et al, 2008)

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joint and separate inversion of strong motion, GPS and DInSAR data;

finite fault is divided into sub-faults;

kinematic parameters are assigned at the grid node and are allowed to vary within a sub-fault;

Kinematic Inversion TechniqueKinematic Inversion Technique

Heat-Bath Simulated Annealing algorithm;

Model Ensemble Ω = Rupture Models m &

Cost Function C(m)

Output of kinematic inversion:

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Inverted Parameters:•Peak Slip Velocity;•Rise Time;•Rupture Time;•Rake.

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Kinematic Inversion:

2007 Niigata-ken Chuetsu-oki Earthquake, Mw=6.6

13 stations (surface or borehole strong motion records);

15 GPS stations ; frequency-band: (0.0÷0.5)Hz; 60 sec (body & surface waves);

all kinematic parameters are

inverted simultaneously.

‘Niigata Data’

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Cost function for the inversion not using KK data

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4% above min cost – 8561 models

min cost - best fitting model

3 % above min cost – 4897 models2% above min cost – 1534 models 1% above min cost – 334 models

0.5% above min cost – 122 models

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Different rupture models can fit the observed data equally well, due to the

intrinsic non-uniqueness of the inversion.

Waveforms fit for the 4%

model ensemble are not obviously

worse than for the 1%

ensemble.

5% SA

Bias (epistemic error)

Aleatory error – Nonunique prediction scatter

Biases at 2-3 s bigger than nonunique prediction scatter

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Bias as the ratio of the observed response

spectrum to the synthetic as a function of period. quantifies the scatter

in the predicted spectral

accelerationForm of aleatory

uncertainty in the ground

motion prediction due to unknown

aspects of slip

distributions.

(data from future EQs will not reduce the uncertainty of this

ground motion prediction

for this past earthquake)

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22%

The scatter in the predicted horizontal

response spectrum at 1g1, from the inversion, is 22% bigger than the scatter at the

Niigata stations.

These results are unaffected by the characteristic of the 1g1 data (not

used in the inversion!)

Comparison of nonunique prediction scatter with aleatory sigma from kinematic and dynamic rupture

simulations of ground motions

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A&S 2008Intra-event

A&S 2008inter-event scatter in the

predictions due to the non-

uniqueness of ‘good’ rupture

models

Variability of predicted ground motion for a M6.5, by considering uncertainties on EGF’ event & variability of

source properties

aleatory variability in ground motions calculated for

kinematic rupture models and theoretical Green's

functions

simulate a M6.5 strike-slip and a M6.5 dip-slip quake at different sites, by adopting

different slip distributions and different hypocenters.

generate source variability by using 2 groups of randomly

variable initial stress distributions, input into

dynamic rupture simulations to generate variable slip models, with fixed and

variable hypo, from which ground motions at different

sites were generated.

aleatory scatter of ground motions from kinematic modeling exceed the

empirically observed scatter in response spectra (blu dashed line)Kinematic rupture models contain too much source variability (real

earthquake ruptures have correlated source properties )

our non-uniqueness scatter at 1g1 for the 4% model ensemble approaches the Abrahamson and Silva’s observed inter-event scatter. This means that the non-

uniqueness variability of predicted motion at 1g1 is almost as great as the variability of motion incurred by using a different earthquake having the same magnitude and

hypocenter to predict the motion at 1g1.

Aleatory variation of Ripperger’s ground motion is smaller than that of kinematic modeling, indicating that the correlations of source properties in dynamic models

do result in lower ground motion variability

Conclusions We computed the scatter in predictions of SA at 1g1 based on the nonuniqueness of inverted kinematic rupture models of the 2007 Niigata-ken Chuetsu-oki earthquake.

The scatter in the predicted horizontal response spectra at 1g1 from the inversion is about 22% bigger than the scatter of response spectra predicted at the Niigata stations. Assuming that this result can be applied to other inversion, in the case of a true interpolation of a ground motion where there is no observation, the scatter in the horizontal component can be 20-25% bigger than the scatter in the motions predicted from good fitting models at sites where there were observed motions.

We compared our nonuniqueness scatter with four different research group estimates of aleatory variability caused by earthquake source properties in kinematic and dynamic rupture modeling: aleatory scatter values of the ground motions from kinematic rupture models exceed the empirically observed scatter in response spectra from a standard GMPE (Abrahamson and Silva, 2008), excluding the effects of variable soil amplification. Clearly these kinematic rupture models contain too much source variability;

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the aleatory variation of dynamic rupture ground motions is much smaller than that of the kinematic rupture models, indicating that the correlations of source properties in dynamic models result in lower ground motion variability;

our non-uniqueness scatter at 1g1 approaches A&S’s observed interevent scatter; the variability of predicted motion at 1g1 caused by nonuniqueness in the rupture model is almost as great as the variability of motion incurred by using a different earthquake having the same magnitude and hypocenter to predict the motion at 1g1.

The nonunique prediction scatter is a hitherto uninvestigated source of aleatory variability that should be included in studies of aleatory variability which use observed slip distributions with randomized hypocenters.

It is not clear whether the non-unique prediction scatter should be added to the other sources of variability in the summarized studies, because none of those studies used a rupture model inferred for a real earthquake. But, if one of the random slip models agrees with a particular earthquake’s rupture model, it would be clear that the non-unique prediction scatter should be added to the aleatory variability.

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Spudich P. and A. Cirella., “Aleatory and Epistemic Uncertainties in Interpolated Ground Motions- Example from the Kasiwasaki-Kariwa Nuclear Power Plant recordings of the July 16, 2007 Niigata-ken Chuetsu-oki, Japan, Earthquake”, , submitted to Bulletin of the Seismological Society of America.

• Accelerograms from the KKNPP were provided by the Association of Earthquake Disaster Prevention, Tokyo, Japan, which is the sole distributor of such data. Copyright of all KKNPP data belongs to the Tokyo Electric Power Company.

• We thank Dr. Kenishi Tsuda of Shimizu Corporation for providing insight into sources of information for this study, and we thank Drs. Arben Pitarka and Enrico Priolo for providing unpublished supplementary materials.

• This work was supported by the U.S. Nuclear Regulatory Commission (NRC) Office of Research Job Code N6501.

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Abrahamson N, Silva W., 2008. Summary of the Abrahamson & Silva NGA Ground-Motion Relations, Earthquake Spectra, 24, No 1., 67-97. Pavic. R., M. Koller, P-Y. Bard, and C. Lacave-Lachet, 2000. Ground-motion prediction with the empirical Green function technique: an assessment of uncertainties and confidence level, J. Seismol., 4, 59-77. Piatanesi, A., A. Cirella, P. Spudich, and M. Cocco, 2007. A global search inversion for earthquake kinematic rupture history: Application to the 2000 western Tottori, Japan earthquake, J. Geophys. Res., 112, B07314, doi:10.1029/2006JB004821. Pitarka, A., P. Somerville and N. Collins, 2002. Numerical simulations for evaluation of median and upper limit ground motions in Switzerland, Unpublished report, URS Corporation, Pasadena.Priolo, E., A. Vuan, P. Klinc, and G. Laurenzano, 2003. Estimation of the median, near fault ground motion in Switzerland, Final report N. 8, PEGASOS Project, Rel. OGS- 33/2003/CRS-4. Ripperger, J., P.M. Mai, and J.-P. Ampuero, 2008. Variability of near-field ground motion from dynamic earthquake rupture simulations,

Bull. Seismol. Soc. Am., 98, 1207-1228. Stark, P.B. and R.L. Parker, 1995. Bounded-variable least-squares: an algorithm and applications. Comp. Stat., 10, 129-141.

Acknowledgements & References

THE END

Supplementary MaterialSupplementary Material

. Modelli Invertiti con & senza kkp con relativi fit figure 7 e 8. Confronto valori di cost function: figure 9. Formule SA(T=2sec); . Formule scatter, bias.. Confronto con gli altri lavori: figure necessarie


random model m0random model m0STARTSTART

++==Strong Strong motionmotion L1+L2 L1+L2 normnorm

To quantify the To quantify the misfit…misfit…

GPSGPSL2 L2 normnorm

C(m)C(m)

DInSARDInSARL2 L2 normnorm

++

Forward Modeling: CompsynForward Modeling: CompsynMisfit computationMisfit computation

Loop over parameters (Vr,…)Loop over parameters (Vr,…)Loop over model valuesLoop over model values

Loop over iterationsLoop over iterationsLoop over temperaturesLoop over temperatures

endendendend

endendendend

Simulated Simulated annealingannealingHeat-bath Heat-bath algorithmalgorithm

Kinematic Inversion TechniqueKinematic Inversion TechniqueStage I: Stage I: Building Model Ensemble-HB Building Model Ensemble-HB

Simulated AnnealingSimulated Annealing

Kinematic Inversion TechniqueKinematic Inversion TechniqueStage I: Stage I: Building Model Ensemble-HB Building Model Ensemble-HB

Simulated AnnealingSimulated Annealing


Best Model Best Model

Average Model:Average Model:

Standard Deviation:Standard Deviation:

Output of kinematic Output of kinematic inversion:inversion:

ΩΩ

Model EnsembleModel Ensemble ΩΩ = = Rupture Models m Rupture Models m &&

Cost Function Cost Function C(m)C(m)

Kinematic Inversion TechniqueKinematic Inversion TechniqueStage II: Stage II: Appraisal of the EnsembleAppraisal of the EnsembleKinematic Inversion TechniqueKinematic Inversion Technique

Stage II: Stage II: Appraisal of the EnsembleAppraisal of the Ensemble


Kinematic Inversion:

2007 Niigata-ken Chuetsu-oki Earthquake, Mw=6.6

13 stations (surface or borehole strong motion records);

15 GPS stations ; frequency-band: (0.02÷0.5)Hz; 60 sec (body & surface waves);

W=31.5km; L=38.5km; =3.5km;

all kinematic parameters are

inverted simultaneously.

South-East Dipping South-East Dipping Fault;Fault;

July 16-th 2007

Best ModelBest Model: : E(m)=0.3E(m)=0.3 Average ModelAverage Model: : E(m)=0.4E(m)=0.4

The ensemble-averaged model better represents the The ensemble-averaged model better represents the stable stable features of the rupture history.features of the rupture history.

• The best fitting model might contain an extreme configuration of model parameters;

• The best fitting model might contain an extreme configuration of model parameters;

Strong MotionStrong Motion

GPSGPS

Comparison between synthetics Comparison between synthetics (red) and observed (blue) (red) and observed (blue) waveforms.waveforms.

Comparison between synthetics Comparison between synthetics (white) and observed (red) (white) and observed (red) coseismic horizontal coseismic horizontal displacement.displacement.


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Formula SA


Comparison of nonunique prediction scatter with aleatory sigma from kinematic and dynamic rupture

simulations of ground motions

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Results1.1. Almost all of the aleatory scatter values of the ground motions from kinematic rupture Almost all of the aleatory scatter values of the ground motions from kinematic rupture

models exceed the blue dashed line, which is the empirically observed scatter in response models exceed the blue dashed line, which is the empirically observed scatter in response spectra, excluding the effects of variable soil amplification. spectra, excluding the effects of variable soil amplification. Clearly these kinematic Clearly these kinematic rupture models contain too much source variabilityrupture models contain too much source variability. Abrahamson (personal communication) . Abrahamson (personal communication) has speculated that real earthquake ruptures have correlated source properties (for has speculated that real earthquake ruptures have correlated source properties (for example, local rupture velocity and peak slip velocity at the same point, for example) example, local rupture velocity and peak slip velocity at the same point, for example) that are not present in the kinematic rupture models, leading to excessive variation in that are not present in the kinematic rupture models, leading to excessive variation in the kinematic rupture models and resulting ground motion. the kinematic rupture models and resulting ground motion.

2.2. The aleatory variation of Ripperger’s ground motions is much smaller than that of the The aleatory variation of Ripperger’s ground motions is much smaller than that of the kinematic rupture models, indicating that the correlations of source properties in kinematic rupture models, indicating that the correlations of source properties in dynamic models do result in lower ground motion variability, although we must bear in dynamic models do result in lower ground motion variability, although we must bear in mind that Ripperger’s variability of stress drop is less than a factor of two, in other mind that Ripperger’s variability of stress drop is less than a factor of two, in other words, rather low. words, rather low.

3.3. Ripperger’s variability is comparable to the empirical variability. Abrahamson and Ripperger’s variability is comparable to the empirical variability. Abrahamson and Silva’s blue dashed line includes the effects of variable source properties (for Silva’s blue dashed line includes the effects of variable source properties (for example, stress drop) as well as variations like directivity amplification that vary from example, stress drop) as well as variations like directivity amplification that vary from site to site in a single earthquake. Ripperger’s variability for variable hypocenter site to site in a single earthquake. Ripperger’s variability for variable hypocenter location agrees rather well with the blue dashed line. However, Ripperger’s result for location agrees rather well with the blue dashed line. However, Ripperger’s result for fixed hypocenter is a bit bigger than the empirically observed earthquake-to-earthquake fixed hypocenter is a bit bigger than the empirically observed earthquake-to-earthquake variation (red dashed line), despite Ripperger’s rather small stress drop variation. variation (red dashed line), despite Ripperger’s rather small stress drop variation.

4.4. Our nonuniqueness scatter at 1g1 for the 4% model ensemble approaches Abrahamson and Our nonuniqueness scatter at 1g1 for the 4% model ensemble approaches Abrahamson and Silva’s observed interevent scatter. This means that the nonuniqueness variability of Silva’s observed interevent scatter. This means that the nonuniqueness variability of predicted motion at 1g1 is almost as great as the variability of motion incurred by using predicted motion at 1g1 is almost as great as the variability of motion incurred by using a different earthquake having the same magnitude and hypocenter to predict the motion at a different earthquake having the same magnitude and hypocenter to predict the motion at 1g1. (We make the ‘same hypocenter’ restriction because differences in directivity 1g1. (We make the ‘same hypocenter’ restriction because differences in directivity contribute to the intraevent error that is characterized by the ‘total’ sigma curve in contribute to the intraevent error that is characterized by the ‘total’ sigma curve in the figure.) Of course, this observation depends on the choice of the 4% ensemblethe figure.) Of course, this observation depends on the choice of the 4% ensemble which which is arbitrary. is arbitrary.

5.5. Spectral acceleration predictions based on ground motion inversions and dynamic Spectral acceleration predictions based on ground motion inversions and dynamic simulations might be less accurate than simply predicting the spectral acceleration from simulations might be less accurate than simply predicting the spectral acceleration from a GMPE such as Abrahamson and Silva (2008) with a directivity correction (for example, a GMPE such as Abrahamson and Silva (2008) with a directivity correction (for example, Somerville et al., 1997). (Or maybe the GMPE Somerville et al., 1997). (Or maybe the GMPE sigmasigma is too small?) is too small?)
