Bulletin of the Seismological Society of America, Vol. 101, No. 5, pp. 2431–2452, October 2011, doi: 10.1785/0120110028
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion by Stephen Hartzell, Arthur Frankel, Pengcheng Liu, Yuehua Zeng, and Sharifur Rahman
Abstract
Four independent ground-motion simulation codes are used to model the strong ground motion for three earthquakes: 1994 Mw 6.7 Northridge, 1989 Mw 6.9 Loma Prieta, and 1999 Mw 7.5 Izmit. These 12 sets of synthetics are used to make estimates of the variability in ground-motion predictions. In addition, ground-motion predictions over a grid of sites are used to estimate parametric uncertainty for changes in rupture velocity. We find that the combined model uncertainty and random variability of the simulations is in the same range as the variability of regional empirical ground-motion data sets. The majority of the standard deviations lie between 0.5 and 0.7 natural-log units for response spectra and 0.5 and 0.8 for Fourier spectra. The estimate of model epistemic uncertainty, based on the different model predictions, lies between 0.2 and 0.4, which is about one-half of the estimates for the standard deviation of the combined model uncertainty and random variability. Parametric uncertainty, based on variation of just the average rupture velocity, is shown to be consistent in amplitude with previous estimates, showing percentage changes in ground motion from 50% to 300% when rupture velocity changes from 2.5 to 2:9 km=s. In addition, there is some evidence that mean biases can be reduced by averaging ground-motion estimates from different methods.
Introduction An important objective for seismologists is to supply engineers with accurate estimates of strong ground motion for future moderate-to-large damaging earthquakes. These ground-motion estimates have evolved over recent years from single parameters such as peak ground acceleration (PGA) to complete time histories of ground motion. Synthetic seismograms are particularly needed to supplement sparse recorded ground motions within 10 km of a fault and for earthquakes of magnitude 7 and larger and to provide for a means of exploring different scenario events with alternative mechanisms and other source parameters. In addition, performing nonlinear analysis in the seismic design of important structures requires the full time series, which must accurately represent potential ground motions in terms of amplitude, duration, frequency content, and the proper phasing of seismic arrivals and pulselike waveforms from rupture directivity. Numerous studies have considered the calculation of synthetic seismograms (Herrero and Bernard, 1994; Hutchings, 1994; Irikura and Kamae, 1994; Zeng et al., 1994; Beresnev and Atkinson, 1997; Hartzell et al., 1999, 2005; Hisada, 2000, 2001; Pitarka et al., 2000; Archuleta et al., 2003; Guatteri et al., 2003, 2004; Liu et al., 2006; Ma et al., 2008; Rezaeian and Der Kiureghian 2008, 2010; Frankel, 2009; Graves and Pitarka, 2010; Mai et al., 2010; among others). In this paper, we consider four of these
approaches, Zeng et al. (1994), Hartzell et al. (2005), Liu et al. (2006), and Frankel (2009), that are examples of the class of source-based, kinematic modeling methods and their associated model and parametric uncertainties. “Sourcebased” refers to the fact that they endeavor to utilize our knowledge of the physical evolution of an earthquake rupture. The models we consider may also use general source relationships observed in dynamic modeling experiments, but which are implemented as kinematic descriptions. Furthermore, the four methods we consider use theoretical Green’s functions. By using Green’s functions from a wave propagation code we are assured of obtaining synthetic seismograms with the expected timing of seismic phases seen in actual ground-motion records, with body waves followed by longer-period surface waves. Accurate directivity effects, leading to coherent pulselike waveforms, are also produced. An important step in embracing synthetic seismograms for design analysis is the development of estimates of their model-based and parametric-based variances. As the partial reference list in the previous paragraph indicates, many methods have been proposed, but we have fewer measures of how the results of these different approaches deviate from one another. We also have few quantitative measures of the sensitivity of the calculated ground motion to the source parameters describing the earthquake rupture process. This paper
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2432 attempts to addresses some of these deficiencies. The variability of observed ground motion is an important quantity in probabilistic seismic hazard assessment (PSHA), and how the variability of observed ground motion compares with that of synthetic seismograms is also of importance. The standard deviation of residuals about a median ground-motion prediction equation is often referred to as sigma (Strasser et al., 2009). As sigma increases, the probability of exceeding a given ground-motion level also increases, particularly at low annual rates of exceedance (Bommer and Abrahamson, 2006). The variability of ground motion in synthetic seismograms is equally important for the evaluation how reliably they reproduce earthquake ground motion. Abrahamson et al. (1990) proposed decomposition of the variability in synthetic ground motion into (1) modeling: adequacy of the form of the model to mimic the earthquake source and wave propagation in the earth; (2) parametric: uncertainty in the input values to the model for future earthquakes; and (3) random: event-toevent and station-to-station variability not accounted for by the model. Modeling and parametric uncertainties are epistemic (uncertainty due to inaccuracies of the model and our incomplete knowledge and limited database), whereas random variability is aleatoric (variability due to random aspects of a natural process). As our knowledge of the earthquake source and propagation effects grows, aspects of the aleatoric variability can be quantified and moved to epistemic uncertainty (Al Atik et al., 2010). Comparison of synthetic seismograms with the recorded ground motion for past earthquakes gives an estimate of the combined model uncertainty and random variability. Parametric uncertainty is estimated by varying the input parameters to the model and making assumptions about their probability distributions. Previous studies have considered aspects of the variability in synthetic ground motions. Ameri et al. (2009) considered two synthetic seismogram methods and obtained approximately a factor of 2 difference in sigma values when comparing synthetic and recorded ground motions. Previous estimates of sigma for the combined model uncertainty and random variability range from about 0.3 to 1.0 natural-log units depending on the frequency and the earthquake data set (Hartzell et al., 2005; Liu et al., 2006; Zeng, 2006; Graves and Pitarka, 2010). In addition, several studies have addressed the variability in synthetic ground motions due to parametric uncertainty by varying the input model parameters (Abrahamson et al., 1990; Pavic et al., 2000; Hartzell et al., 2005; Hutchings et al., 2007; Sorensen et al., 2007; Ripperger et al., 2008; Wang et al., 2008; Causse et al., 2008; Beauval et al., 2009; among others). These studies yield estimates of parametric uncertainty with a range in sigma from about 0.35 to 1.1 natural-log units, depending on the frequency band and the parameters considered. These values are generally, for a single source region, observed at multiple stations. However, these estimates are difficult to evaluate because of the often subjective nature of the selection process for the allowable range in model parameters. We can compare these ranges in sigma with values reported for
S. Hartzell, A. Frankel, P. Liu, Y. Zeng, and S. Rahman
observed ground motions. Typical sigma values for regional empirical ground-motion data sets range from 0.5 to 0.8 natural-log units (Abrahamson and Silva, 1997; 2008; Boore et al., 1997; Boore and Atkinson, 2008; Strasser et al., 2009). However, smaller values are obtained considering singlestation variability due to regional sources, 0.60 to 0.67 (Atkinson, 2006); or variability at a single site from a single source zone, 0.41 to 0.46 (Atkinson, 2006), and similarly 0.34 to 0.46 (Morikawa et al., 2008). Continuing this trend of specificity of the source and receiver, Anderson et al. (2000) estimated a sigma of 0.3 natural-log units for repeating large characteristic earthquakes observed at a single site. Consistent with these observations, Lin et al. (2011) found a 9%–14% reduction in sigma when considering a single site and multiple sources and a 39%–47% reduction for fixed source and site locations. This trend toward lower sigma by the consideration of specific source, path, and site effects, rather than applying a more globally collected ground-motion database, is referred to as nonergodic PSHA (Anderson and Brune, 1999; Walling, 2009). In this paper, we explore the range in sigma for combined model uncertainty and random variability through the modeling of observed ground motions. By using four different modeling methodologies, we consider variances averaged over different sources for a given method, as well as variances averaged over different methods for a given source. We further consider the model uncertainty alone by comparing synthetic ground motions to the average of the four methods, independent of the recorded ground motions. Finally, we consider the sensitivity of the synthetic ground motion to a single important source parameter, the rupture velocity. We first summarize the four methodologies.
Source-Based Kinematic Methods Summary characteristics of the four modeling methods considered are listed in Table 1. More detailed information is given in the following sections. Frankel Frankel (2009) presents a method to compute synthetic seismograms using a source with constant stress-drop scaling with magnitude. The procedure combines long-period synthetics made using Green’s functions for a prescribed velocity model with short-period synthetics made using point-source stochastic seismograms from the procedure of Boore (1983). The point-source stochastic synthetics are convolved with a function Sf C1 f=f0small 2 = 1 f=f0main 2 (Frankel, 1995) that ensures that the resulting acceleration spectrum is flat for frequencies lower that the corner frequency of the point-source seismograms. The long-period and short-period synthetics are combined using a matched filter (Hartzell and Heaton, 1995), where the transition frequency varies with magnitude. The frequency of transition is inversely related to observed near-fault velocity
1D full wave theory
Liu et al. (2006)
Zeng et al. (1994)
*Finn et al. (1975); Lee and Finn (1978) † Bonilla et al. (1998); Bonilla (2000)
1D full wave theory
Hartzell et al. (2005)
Green’s Function
1D full wave theory at low frequency, stochastic at high frequency 1D full wave theory
Frankel (2009)
Synthetic Model
Table 1
Fractal size distribution; overlapping sources
Uniform source grid; nonoverlapping sources
Fractal size distribution; nonoverlapping sources
Uniform source grid; nonoverlapping sources
Source Distribution
κ2 , with x and y correlation lengths; based on uniform random number distribution κ2 , with adjustable corner wavelength; based on uniform random number distribution κ2 , with x and y correlation lengths; based on Cauchy random number distribution Based on fractal size distribution and Brune (1970, 1971) model
Slip Distribution
Brune (1970, 1971)
Functional form from dynamic modeling
Brune (1970, 1971)
Brune (1970, 1971)
Subevent Source-Time Function
Synthetic Ground-Motion Models Used in This Study
Secant rupture velocity with random subevent initiation points Secant rupture velocity correlated with slip amplitude and rise time Secant rupture velocity with random subevent initiation points
Secant rupture velocity with randomness
Rupture Velocity
DESRA2†
NOAHW†
NOAHW†
DESRA2*
Local Site and Soil Nonlinearity
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion 2433
2434 pulse durations. An extended fault plane with area Amain is divided into subevents such that the subevent area, Asub Amain M0sub =M0main 2=3 , where M0 is seismic moment based on constant stress-drop scaling. The slip on each subevent follows a κ2 distribution, where κ is the spatial wavenumber of the slip, consistent with constant stress-drop scaling with seismic moment (Andrews, 1980, 1981). The spatial correlation of slip (inversely proportional to corner wavenumber) along strike and down dip follows the magnitude-dependent values of Mai and Beroza (2002). The source-time function of the slip is the Brune (1970, 1971) pulse with a corner frequency equal to the reciprocal of the rise time. The rise time for each subevent, T r M0sub = μA0sub Sv u=Sv , where μ is fault-zone rigidity, u is slip, and Sv is a specified average slip velocity. The rupture velocity may be chosen to follow either a local rupture-velocity scheme or a secant rupture velocity. The secant model calculates rupture times by taking the distance from the hypocenter to the subfault location divided by the rupture velocity and generally introduces greater variation in rupture times. Random components are added to the rupture time, strike, dip, and rake to simulate heterogeneity in the earth. Hartzell The synthetic seismogram model of Hartzell et al. (2005) is based on a fractal distribution of sources (Irikura and Kamae, 1994), where the smallest subevent size is determined using the constant stress-drop scaling model, Asmallest Amain M0smallest =M0main 2=3 , for a prescribed M0smallest . The response of each subevent is calculated by summing theoretical Green’s functions for a given velocity model for all frequencies. The source size distribution has a fractal dimension of 2 (number of subevents of a given size inversely proportional to their area), which for nonoverlapping subevents yields an ω2 high-frequency spectral falloff (Frankel, 1991). For L subevent sizes, the number of subevents of area Ak is Amain =LAk . The corner frequency of the smallest subevent is given by fsmallest 4:9 × 106 βΔσ=M0smallest 1=3 (Brune, 1970, 1971), where Δσ is a prescribed stress drop. The Brune (1970, 1971) source-time function is applied to the smallest subevent with rise time equal to the reciprocal of the corner frequency plus a random term. The larger subevents are constructed by summing the required number of the smallest subevent and convolved with the Frankel (1995) operator, Sf C1 f=f0smallest 2 =1 f=f0larger 2 , that ensures that the acceleration spectra of the fractal subevents is flat below the corner frequency of the smallest subevent. Corner frequencies for the different-sized subevents are determined using the ω2 spectral scaling relationship fsmallest =flarger M0larger =M0smallest 1=3 . The slip distribution is the κ2 spatial wavenumber model with the same corner wavenumber along the strike and down the dip, which leads to slip values that appear equally random from every point on the fault. Both local and secant rupture-velocity timing are implemented in addition to optional random loca-
S. Hartzell, A. Frankel, P. Liu, Y. Zeng, and S. Rahman
tions for subevent hypocenters rather than the closest point to the main event hypocenter (Zeng, 1994). This later option can be used to regulate the amount of rupture directivity. Random terms are added to the rupture time and rake. Liu The synthetic seismogram method of Liu et al. (2006) utilizes the concept of correlated random source parameters and theoretical wave-propagation Green’s functions at all frequencies. Dynamic modeling of earthquake ruptures (Guatteri et al., 2003) has shown that areas of large slip correlate with faster rupture velocity. Larger slip is also expected to require longer rise times. In this model, slip, rupture velocity, and rise time are represented as random, correlated variables with adjustable correlation coefficients. Three 2D Gaussian random fields are generated with the power spectral density of Mai and Beroza (2002), Pkx ; ky 1=1 kx CL 2 ky CW 2 2 , where CL and CW are coherence lengths along the strike of the fault and down the dip. Each of these distributions is mapped into spatial distributions of source parameters that have designated probability distributions as follows (Cario and Nelson, 1997): (1) slip: Cauchy probability distribution allowing for greater variability in slip; (2) secant rupture velocity: uniform distribution between 0.6 and 1.0 times the local shear-wave velocity; and (3) rise time: beta distribution allowing for fixed upper and lower bounds. These 2D distributions are then used in a subevent summation process. The source-time function for each subevent is motivated by dynamic faulting results (Guatteri et al., 2004), with a sharp onset and slower decay. Random factors are also added to the strike, dip, and rake. Zeng In the synthetic seismogram method of Zeng et al. (1994), a fractal distribution of source sizes is used with overlapping sources and wave propagation Green’s functions for a specified velocity model. Subevents are distributed randomly on the fault plane with a size distribution based on the self-similar model proposed by Frankel (1991), where the number of subevents with radii larger than R is NR C=DRD RD max and D 2 is the fractal dimension, Rmax is the radius of the largest subevent, and C is a proportionality constant. The stress drop of the subevent, Δσ, is assumed to be independent of the subevent radius and related to the subevent moment by M0 R 16=7R3 Δσ (Keilis Borok, 1959). The source-time function is the Brune (1970, 1971) pulse. The corner frequency of each subfault is related to the source radius by fc 2:34β=2πR (Brune, 1970, 1971). Secant rupture velocity timing is used with the ability to implement a random hypocenter for each subevent rather than the center point of the subevent. This option can be used to reduce rupture directivity, if required. Effects of scattering are added to model propagation through a heterogeneous structure by adding an isotropic scattering term (Aki and Cheout, 1975; Zeng et al., 1995).
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion
2435
Table 2 Northridge Strong-Motion Stations Station Code
Station Name
Latitude
Longitude
NEHRP Site Class
BHG CPP ECC ER1 GRI HOL HWS JEM JFP KAG LA0 LA1 LA2 LA3 LA4 LAW MPP NHC NWH NWP PAR PDM RRS RSE SFP SMC SMP SSC SSP SYF TAG TMP TPC UCL VAN VER WC1 WIL WOO WVN WVS WVU
Baldwin Hills Canoga Park, 7769 Topanga Canyon No. 53 Energy Control Center-Grnd Flr Encino Reservoir Dam Griffith Park Obs. Van Nuys Hotel Los Angeles, Faring Rd. No. 6 Jensen Filter Plant Generator Building Jensen Filter Plant Administration Bld. Pacoima, Kagel Canyon Fire Station Stone Canyon Reservoir Beverly Hills, Mulholland Dr. Century City Hollywood Storage Bld. Alhambra, Fremont School Los Angeles, 8510 Wonderland Ave. Moorpark North Hollywood, Coldwater Cn. Newhall, 26835 Pico Cn. Blvd. Newhall, Los Angeles County Fire Dept. Santa Clarita, Pardee Substation Pacoima Dam, Downstream Sylmar, Rinaldi Receiving Station San Fernando, Receiving Station East Arleta, Nordhoff Ave. Fire Station Santa Monica City Hall Simi Valley, 6334 Katherine No. 55 Los Angeles, Saturn St. No. 91 Santa Susana, Dept. of Energy Sylmar County Hospital Tarzana Nursery Los Angeles, Temple of Hope Topanga Fire Station UCLA Grounds Sepulveda VA Hospital Los Angeles, 3620 S. Vermont Ave. White Oak Church No. 003 1100 Wilshire Blvd. Wood Ranch Reservoir Wadsworth VA Hospital, North Wadsworth VA Hospital, South Los Angeles, Grand Ave No. 22
34.0086 34.212 34.259 34.1488 34.118 34.2209 34.0893 34.313 34.312 34.288 34.1062 34.1317 34.063 34.09 34.07 34.1144 34.2871 34.194 34.391 34.388 34.435 34.3341 34.281 34.176 34.2369 34.0122 34.2632 34.0467 34.2309 34.326 34.1604 34.059 34.084 34.068 34.2493 34.022 34.2087 34.0521 34.24 34.0543 34.0499 34.005
118:3615 118:605 118:336 118:5154 118:299 118:4708 118:433 118:498 118:496 118:375 118:4542 118:4394 118:418 118:339 118:15 118:3797 118:8816 118:412 118:622 118:5332 118:582 118:3979 118:479 118:36 118:4391 118:4914 118:6673 118:3556 118:7135 118:444 118:5343 118:246 118:599 118:439 118:4777 118:2925 118:5174 118:2629 118:82 118:4531 118:4485 118:279
CD D C BC B D C C C C C C CD D C BC C C D D D BC D D D CD C D BC CD CD CD CD CD D D D CD C C D C
Nonlinear Soil Correction In this study, each of the four synthetic seismogram methods described in the previous sections applies one of two different nonlinear soil corrections to account for sitespecific nonlinearity. Codes by Hartzell et al. (2005) and Liu et al. (2006) use NOAHW (Bonilla et al., 1998; Bonilla, 2000) and codes by Frankel (2009) and Zeng et al. (1994) use DESRA2 (Finn et al., 1975; Lee and Finn, 1978). Two different codes are used in an attempt to more completely represent model uncertainty; however, these assignments are arbitrary. In our tests, both codes yielded similar results so a different assignment would produce similar ground motions. Both codes calculate the fully nonlinear seismic response of a stack of horizontal soil layers to a vertically
incident shear wave based on the hyperbolic stress-strain model described by Konder and Zelasko (1963). We assume total stress or dry conditions. The National Earthquake Hazards Reduction Program (NEHRP; 1994) soil classification scheme is used to consider the effects of different soil types. Site categories are based on the average shear-wave velocity in the top 30 m (V S30 ): A (> 1500 m=s), B (760 to 1500 m=s), C (360 to 760 m=s), D (180 to 360 m=s), and E (< 180 m=s). Our nonlinear corrections use average shear-wave velocity profiles for these site classes as well as the midrange site classes BC and CD developed by Silva et al. (1998). These velocity profiles were developed by selecting velocity profiles with V S30 values closest to the midrange of the NEHRP site categories. These profiles are then averaged and shifted so that their V S30 values agree with
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S. Hartzell, A. Frankel, P. Liu, Y. Zeng, and S. Rahman
Table 3 Loma Prieta Strong-Motion Stations Station Code
Station Name
Latitude
Longitude
NEHRP Site Class
AAR ADD AGH BEL BRN BSH BSW BUE CAC CAF CAL CAP CLA CLD COR CSR ECG FCM FCR FMS FSH GAS GGB GL1 GL3 GL4 GL6 HAP HCH HFS HMS HOL HSO HTS HVG LBL LCD LEX LGP MCH MIB MPH MVH OOH OSR PAL PAO PAV PPJ RBM RCS SAA SAG SAL SBG SBO SCA SCB SCH SFA SFB
APEEL Array 2, Redwood City Anderson Dam Downstream Agnews State Hospital Belmont, two-Story Office Bldg. Bran Berkeley, 2-Story Hospital Berkeley, 2168 Shattuck, W. Basement Yerba Buena Island Calaveras Array, Cherry Flat Calaveras Array, Fremont Calaveras Array, Calaveras Capitola, Fire Station Coyote Lake Dam, Southwest Abutment Coyote Lake Dam, Downstream Corralit APEEL 9, Crystal Springs Reser Emeryville, 6363 Christie Foster City, Menhaden Court Foster City, Redwood Shores Fremont, Mission San Jose S. San Francisco, 4-Story Hospital Upper Crystal Springs Res.—Pulgas Golden Gate Bridge, Abutment Gilroy 1, Gavilan College, Water Tank Gilroy 3, Gilroy Sewage Plant Gilroy 4, San Ysidro School Gilroy 6, San Ysidro Hollister Airport Hollister City Hall Annex, Basement Hayward, Four-Story School Bldg. Hayward, Muir School Hollister, South Street and Pine Drive Hayward, Six-Story Office Bldg. Hayward, 13-Story School Office Bldg. Halls Valley, Grant Park Berkeley, Lawrence Berkeley Lab Lower Crystal Springs Dam Lexington Dam LGPC Monterey, City Hall Milpitas, Two-Story Industrial Bldg. Menlo Park VA Hospital, Bldg. Martinez, VA Hospital, Basement Oakland, Outer Harbor Wharf Oakland, 24-Story Residential Bldg. Palo Alto, Two-Story Office Bldg. Palo Alto, Two-Story Office Bldg. Palo Alto VA Hospital, Bldg. 1 Piedmont, Piedmont Jr. High Grounds Richmond Bulk Mail Cntr., 2501 Redwood City, Three-Story School Office Bldg. Saratoga, Aloha Ave. Sago South, Hollister, Cienega Rd. Salinas, John and Work St. San Bruno, Nine-Story Govt. Office Bldg. San Bruno, Six-Story Office Bldg. Sunnyvale, Colton Ave. San Jose, 10-Story Commercial Bldg. San Francisco, Cliff House San Francisco Int. Airport San Francisco, 18-Story Commercial Bldg.
37.52 37.165 37.397 37.512 36.973 37.855 37.87 37.81 37.396 37.535 37.452 36.974 37.118 37.124 37.046 37.47 37.844 37.555 37.55 37.53 37.66 37.49 37.806 36.973 36.987 37.005 37.026 36.888 36.851 37.657 37.657 36.848 37.635 37.655 37.338 37.876 37.529 37.202 37.172 36.597 37.43 37.468 37.993 37.816 37.798 37.453 37.806 37.4 37.823 37.884 37.448 37.255 36.753 36.671 37.627 37.628 37.402 37.338 37.78 37.622 37.792
122:25 121:631 121:952 122:308 121:995 122:256 122:27 122:36 121:756 121:929 121:807 121:952 121:55 121:551 121:803 122:32 122:295 122:248 122:23 121:919 122:439 122:31 122:472 121:572 121:536 121:522 121:484 121:413 121:402 122:053 122:082 121:397 122:104 122:056 121:714 122:249 122:361 121:949 122:01 121:897 121:897 122:157 122:115 122:314 122:257 122:112 122:267 122:14 122:233 122:302 122:265 122:031 121:396 121:642 122:424 122:424 122:024 121:893 122:51 122:398 122:4
E D D C CD CD C C C CD C C C D C C D E E C D BC C B D D C D D C C C D C D B C B C C D D C D D D D C B D BC C C D CD CD D D C D E (continued)
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion
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Table 3 (Continued) Station Code
SFC SFD SFS SGB SJ3 SJH SJI SKY SMB SPH SPS SPT SRB SRH SSB SSG SSU SUS TRI UCS WAH WAT
Station Name
Calaveras Array, Sunol Fire St. San Francisco, Diamond Heights San Francisco, 1295 Shafter San Jose, 13-Story Govt. Office Bldg. San Jose, Three-Story Office Bldg. San Jose, Santa Teresa Hills San Jose Interchange Upper Crystal Springs Res., Skyline San Francisco, 575 Market, Basement San Francisco, Pacific Heights San Francisco, Presidio S. San Francisco, Sierra Pt. San Jose, 10-Story Residence San Francisco, Rincon Hill San Francisco, Six-Story School Bldg. Saratoga, One-Story School Gym San Francisco State U., Thorn Stanford University, SLAC Treasure Island Santa Cruz, UCSC/Lick Lab WAHO Watsonville, Four-Story Commercial Bldg.
the midrange values for the different NEHRP site categories. Tables 2–4 list the station soil site classifications. In addition to shear-wave velocity information, the degree of nonlinearity within each profile is quantified by the widely used modulus reduction curves developed by the Electric Power Research Institute (EPRI) (1993) for rock and sand and the curves of Vucetic and Dobry (1991) for clay. The EPRI (1993) rock curves are used for site classes B and BC; the sand curves for site classes C, CD, and D; and the Vucetic and Dobry (1991) curves for class E. Further details are given in Hartzell et al. (2004).
Latitude
Longitude
NEHRP Site Class
37.597 37.74 37.728 37.353 37.212 37.21 37.34 37.465 37.79 37.79 37.792 37.674 37.338 37.79 37.762 37.262 37.724 37.419 37.825 37.001 37.047 36.909
121:88 122:43 122:385 121:903 121:803 121:803 121:851 122:343 122:4 122:43 122:457 122:388 121:888 122:39 122:459 122:009 122:475 122:205 122:373 122:06 121:985 121:756
C C B D C C D C E B C B D B D C CD C E B CD D
peak ground velocity (PGV), peak ground displacement (PGD), Fourier and response spectral amplitudes, and integrals of the squared acceleration or velocity (Anderson, 2004; Olsen and Mayhew, 2010). The goodness-of-fit measure of Kristekova et al. (2006) compares the differences in the envelope and phase of synthetic and recorded ground motions. In addition, parameters of more engineering relevance have been suggested, including the ratio between the inelastic and elastic response spectra (Bazzurro et al., 2004; Baker and Jayaram, 2008; Olsen and Mayhew, 2010), and a damage potential index (S. Pei et al., 2011, unpublished manuscript). In this study, we use the criteria applied by
Model Events and Velocity Models Ground-Motion Data Sets and Error Measures Three earthquakes have been chosen for our comparative analysis of the four synthetic seismogram computation methods: 1994 Northridge, California; 1989 Loma Prieta, California; and 1999 Izmit, Turkey. A list of their important source parameters is given in Table 5. Northridge and Loma Prieta are thrust events, and Izmit is a large strike-slip earthquake. These events were selected because each has been well studied and each has good strong-motion station coverage. The strong-motion station locations used in the analysis are listed in Tables 2–4 and plotted in Figure 1a–c. Results for bias and standard deviation relative to recorded ground motion use all the stations in Tables 2–4. If a subset of these stations were used, the results could vary; however, we have attempted to include all records with significant ground motion. Previous discussions of the goodness-of-fit criteria for synthetic seismograms have mainly dealt with averages of various time- and frequency-domain measures, such as PGA,
Table 4 Izmit Strong-Motion Staions Station Code
Station Name
Latitude
Longitude
NEHRP Site Class
ARC ATK BRS BUR CNA DHM DZC GBZ GYN IST IZN IZT MCD MSK SKR YPT ZYT
Arcelik Atakoy Bursa Bursa Cekmece Yesilkoy Duzce Gebze Goynuk Istanbul Iznik Izmit Mecidiyekoy Maslak Sakaraya Yarimca Zeytinburnu
40.824 40.989 40.183 40.261 41.024 40.982 40.844 40.82 40.396 41.058 40.44 40.79 41.065 41.104 40.737 40.764 40.986
29.361 28.849 29.131 29.068 28.759 28.82 31.149 29.44 30.783 29.013 29.75 29.96 28.997 29.019 30.384 29.762 28.908
B D B D B B D B B B D B B B B D D
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S. Hartzell, A. Frankel, P. Liu, Y. Zeng, and S. Rahman
Table 5 Earthquakes Modeled Earthquake
Moment Magnitude
Recording Stations
Strike
Dip
Rake
Fault Length (km)
Fault Width (km)
Depth of Hypocenter (km)
Northridge (1994) Loma Prieta (1989) Izmit (1999)
6.7 6.9 7.5
42 83 17
122 128 90
40 70 90
90 144 180
20 40 160
24 20 17
17.5 18.0 15.9
Hartzell et al. (2005), including the standard-deviation estimate of Schneider et al. (1993), based on original work by Abrahamson et al. (1990). This measure takes the difference of the natural logarithms of the observed Of and calculated Hf response spectra for the geometric average of the two horizontal components,
required by our analysis, only 1D structures have been considered. If 3D velocity models are used, it is expected that fits to the strong-motion records at periods longer than 1 s would improve, particularly in areas with pronounced basin effects (Hartzell et al., 2005; Day et al., 2008). The strongmotion stations for the Northridge and Loma Prieta earth-
v u N u1 X σf t flnO1i f × O2i f1=2 lnH1i f × H2i f1=2 Bfg2 ; N i1
and a model bias given similarly by Bf
N 1X O1i f × O2i f1=2 ; ln N i1 H1i f × H 2i f1=2
(2)
where N is the number of stations. We use these measures for both 5% damped response spectra and Fourier spectra. When averages are calculated over different modeling methodologies for a given earthquake or over different earthquakes for a given model, we calculate the standard deviation of the average of the individual variances. A similar averaging process is used to calculate the standard deviation over a given frequency band. We also use several time-domain measures of the goodness-of-fit because frequency-domain measures give a poor indication of how well peak motions and duration are matched. These measures are the standard deviation and model bias of the geometric average of the two horizontal components taken over all stations for the peak velocity, peak acceleration, cumulative sum of the squared velocity, velocity duration, and acceleration duration. Analogous expressions to equations (1) and (2) are used. The velocity and acceleration durations are given by the time interval between the 5% and 95% levels of the cumulative sum of the squared velocity and acceleration, respectively. Velocity Models and Green’s Functions Each of the synthetic seismogram codes used in this paper sums Green’s functions from a 1D velocity model. These Green’s functions are all calculated using the frequency-wavenumber method of Zhu and Rivera (2002). The codes are not restricted to 1D Green’s functions and could use Green’s functions from a 3D velocity model at lower frequencies. However, to facilitate the numerous runs
(1)
quakes are divided into rock and soil sites. This designation does not strictly pertain to the near-surface velocity, which is reflected in the NEHRP site classification, but in the deeper structure. A soil site here refers to stations located over deeper sedimentary fill, such as the Los Angeles basin or the Santa Clara valley in the Bay Area. A single velocity model is used for all the Izmit earthquake stations. Table 6 lists the seismic properties for each of these velocity models.
Combined Model Uncertainty and Random Variability To estimate the combined model uncertainty and random variability, we simulate the recorded ground motion for three earthquakes using each of the four codes. These codes were run independently of each other. No attempt was made to duplicate input parameters among the codes, which would be difficult given each code’s different parameterization. The only commonalities are the velocity model, moment, and general dimensions and mechanism of the earthquake. Each code’s input parameters were optimized in a trial-and-error process of fitting the observed ground motion. This process most closely tracks what would be obtained by different investigators using their respective codes and working independently of each other. The error measures, discussed earlier, were used to evaluate the goodness of fit. As an example of the variability of the input, Figure 2 compares the slip distributions used by the four methods in the case of the Northridge earthquake. The general character of the slip distributions is controlled mainly by the formulation used in each code, rather than by tuning of user-controlled parameters. The large variability in the slip distributions is not reflected in the resulting ground motions. That is, the ensemble fit to the ground motion is less sensitive
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion
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Figure 1. Maps of strong-motion stations used in the analysis of (a) 1994 Mw 6.7 Northridge earthquake, (b) 1989 Mw 6.9 Loma Prieta earthquake, and (c) 1999 Mw 7.5 Izmit earthquake. Surface projection of the fault plane is indicated by the rectangular region for Northridge and Loma Prieta and the east–west trace for Izmit. Surface NEHRP soil classifications are given for Northridge and Loma Prieta.
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S. Hartzell, A. Frankel, P. Liu, Y. Zeng, and S. Rahman
Table 6 Rock and Soil Velocity Models for Northridge and Loma Prieta, and Velocity Model for Izmit V P (km=s)
V S (km=s)
QP
QS
Rock Velocity Model for Northridge and Loma Prieta 2 0.8 2 0.1 2.5 1.1 2.1 0.4 3 1.5 2.2 0.5 3.8 2 2.3 1 4.7 2.7 2.5 3 5.5 3 2.6 3 6 3.6 2.8 7 6.5 3.9 3 10 7.8 4.5 3.3 -
90 200 300 450 600 1000 1500 2000 2000
60 150 200 300 400 500 750 1000 1000
Soil Velocity Model for Northridge and Loma Prieta 1.5 0.4 1.8 0.1 1.9 0.6 1.9 0.1 2.1 0.8 2 0.3 2.4 1.1 2.1 0.5 3.1 1.5 2.2 1 4.5 2.5 2.5 3 5.5 3 2.6 3 6 3.6 2.8 7 6.5 3.9 3 10 7.8 4.5 3.3 -
30 50 100 200 300 450 1000 1500 2000 2000
20 35 75 150 200 300 500 750 1000 1000
Izmit Velocity Model 2.15 1.25 3.2 1.8 4.69 2.71 5.16 2.98 5.76 3.32 6.12 3.53 6.27 3.62 6.55 3.78 6.86 3.96 7.2 4.15
100 100 200 433 550 700 800 800 1000 1000
50 50 100 217 275 350 400 400 500 500
Density (g/cc)
2.1 2.1 2.43 2.53 2.65 2.74 2.78 2.86 2.94 3.04
to the details of the slip than to the details of the timing of the rupture, as concluded by Hartzell et al. (2005). Figure 3a,b,c compares time-domain records at near-fault stations for the Northridge, Loma Prieta, and Izmit earthquakes, respectively. We have selected these stations to reduce propagation path effects and focus on source effects. Each of the four methods obtains waveforms that differ from one another but that fit the observed records approximately equally well using our spectral and time-domain measures. We next quantify this fit. Bias Figures 4 through 7 show values of bias and standard deviation averaged over different frequency bands for response spectra and Fourier spectra as well as five time-domain measures. All values are in natural-log units. The averages in the last column and row of Figures 4 through 6 are over different modeling methods for a given earthquake and over different earthquakes for a given method, respectively. The bias averages in Figure 7 are averages of the absolute values of the individual biases compared with those in Figures 4 through 6,
Thickness (km)
0.12 0.38 1 3 4 4 12 5 5 -
which include the sign of the biases. The average biases in Figures 4 through 6 are useful for assessing if averaging the ground-motion estimates from several different methods will lead to lower bias values. The results show, with no prior knowledge of which method may perform the best, that taking averages of ground-motion estimates from several different methods can lead to lower biases, particularly for the Northridge and Izmit events. Bias averages over different earthquakes in Figures 4 through 6 point out trends that may exist for a given method, that is, consistently high or consistently low at a given frequency or time-domain measure. No strong trends are observed. From Figure 7, the average of the absolute values of the biases range from near zero to about 0.3 natural-log units. Most of the frequency measures (response and Fourier spectra) lie between 0.2 and 0.3. The worst timedomain bias is for velocity duration with the synthetics consistently smaller than the data. This trend is attributed to the use of 1D synthetics, which fail to fully capture the longerduration surface-wave energy in sedimentary basins. As already stated, 3D synthetics can be used with any of these methods, which could reduce or eliminate this trend.
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion Standard Deviation Most of the values of standard deviation lie between 0.5 and 0.7 natural-log units for response spectra and 0.5 and 0.8 for Fourier spectra. The lower range for response spectra is due to the averaging effect over frequency. Larger values are obtained for the lowest frequency bin for the Loma Prieta earthquake that are clearly related to the processing of the data
Figure 2.
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records, which applied high-pass frequency filters with differing corner frequencies. The effect of this filtering is also seen in the larger negative bias at these frequencies. These values are therefore discounted in our interpretation. The range in standard deviations for individual model/event combinations and for averages over methodologies and earthquakes is approximately the same as seen in observed ground motions
Example slip distributions used by the four different synthetic seismogram codes for the 1994 Northridge earthquake. Absolute scaling of values is different between slip distributions.
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S. Hartzell, A. Frankel, P. Liu, Y. Zeng, and S. Rahman
(Strasser et al., 2009). The time-domain measure with the largest standard deviation is the cumulative sum of the squared velocity. This result is not surprising, because by squaring the velocity, any difference between the observed and synthetic records is amplified. Peak velocity and peak acceleration have lower standard deviations between 0.4 and 0.6 natural-log units, and the duration measures for velocity and acceleration are between 0.2 and 0.4.
Model Epistemic Uncertainty If we look at just the variability between the predictions of the different models without reference to any recorded ground motion, we can obtain an estimate of the model
(a)
Northridge 22
Station JEM
uncertainty. This reasoning assumes that each of the models in the analysis is consistent with our current knowledge base and does not represent an outlier in terms of model predictions. From the results of the previous section, we think these conditions are met. However, because the individual model estimates were fine-tuned to fit the same sets of ground motion, we may underestimate the model uncertainty present in a blind-scenario modeling study. Figure 8a–d shows frequency and time-domain estimates of standard deviation among the four model ground-motion predictions relative to the average of all the model predictions using the same events and stations as the previous analysis. Most of the values of standard deviation lie between 0.2 and 0.4, which is about one-half of our estimates for the standard deviation Loma Prieta
(b) 0
292
Data
Station WAT
90
Data
Hartzell 100
Hartzell 100 c m/sec
Frankel
c m/sec
Frankel
0 Liu
0
Zeng
Zeng
Liu
Seconds
Seconds
Seconds
(c)
Seconds
Izmit
0
Station GBZ
270
Data
Frankel
c m/sec
Hartzell 100
0 Liu
Zeng
Seconds
Seconds
Figure 3. Comparison of horizontal time-domain velocity records from the four different codes at a near-fault station for (a) Northridge earthquake, (b) Loma Prieta earthquake, and (c) Izmit earthquake. Data and synthetic records are bandpass filtered from 0.1 to 10 Hz.
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion
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Response Spectra Bias
(a) Northridge
Frankel
Hartzell
Liu
Zeng
Average
0.2 0 -0.2
Loma Prieta
-0.4
0.2 0 -0.2 -0.4
Izmit
0.2 0 -0.2 -0.4 1 2 3 4 5 6
Average
0.2 0 -0.2 -0.4 1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
Response Spectra Standard Deviation
(b) Northridge
Frankel
Hartzell
Liu
Zeng
Average
0.7 0.6
Loma Prieta
0.5
0.7 0.6 0.5
Izmit
0.7 0.6 0.5 1 2 3 4 5 6
Average
0.7 0.6 0.5 1 2 3 4 5 6
Figure 4.
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
(a) Values of response spectral bias (observed/synthetic) in natural log units for the logarithmically distributed frequency bands: (1) 0.2–0.37 Hz, (2) 0.37–0.71 Hz, (3) 0.71–1.34 Hz, (4) 1.34–2.53 Hz, (5) 2.53–4.77 Hz, (6) 4.77–9.0 Hz. Averages in the far right column and the last row are over different simulation methods for a single earthquake and over different earthquakes for a single method, respectively. (b) Response spectral standard deviation calculated by Equation (1) in natural-log units for the same frequency bands as in (a).
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S. Hartzell, A. Frankel, P. Liu, Y. Zeng, and S. Rahman Fourier Spectra Bias
(a) Hartzell
Frankel
Liu
Zeng
Average
Northridge
0.4 0.2 0 -0.2 -0.4
Loma Prieta
0.4 0.2 0 -0.2 -0.4 0.4
Izmit
0.2 0 -0.2 -0.4 1 2 3 4 5 6 0.4
Average
0.2 0 -0.2 -0.4 1 2 3 4 5 6
1 2 3 4 5 6
Frankel
Hartzell
1 2 3 4 5 6
1 2 3 4 5 6
Fourier Spectra Standard Deviation
(b)
Liu
Zeng
Average
1 0.9 0.8 0.7 0.6 0.5 1 0.9 0.8 0.7 0.6 0.5 1 0.9 0.8 0.7 0.6 0.5 1 2 3 4 5 6 1 0.9 0.8 0.7 0.6 0.5 1 2 3 4 5 6
1 2 3 4 5 6
Figure 5.
1 2 3 4 5 6
1 2 3 4 5 6
Same as Figure 4 but for Fourier spectra.
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion
(a)
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Time Domain Parameter Bias
Northridge
Frankel
Hartzell
Liu
Zeng
Average
0.4 0.2 0
Loma Prieta
-0.2
0.4 0.2 0 -0.2
Izmit
0.4 0.2 0 -0.2 1 2 3 4 5
Average
0.4 0.2 0 -0.2 1 2 3 4 5
(b)
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
Time Domain Parameter Standard Deviation Frankel
Hartzell
Liu
Zeng
Average
Northridge
1 0.8 0.6 0.4 0.2
Loma Prieta
1 0.8 0.6 0.4 0.2 1
Izmit
0.8 0.6 0.4 0.2 1 2 3 4 5
Average
1 0.8 0.6 0.4 0.2 1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
Figure 6. (a) Time-domain parameter bias (observed/synthetic) in natural-log units for the parameters (1) peak velocity, (2) peak acceleration, (3) cumulative squared velocity, (4) velocity duration, and (5) acceleration duration. Averages are the same as in Figure 4. (b) Timedomain parameter standard deviation in natural-log units for the same parameters as in (a).
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S. Hartzell, A. Frankel, P. Liu, Y. Zeng, and S. Rahman Absolute Value Averages of Bias
(a)
Response Spectra
Fourier Spectra
Time Domain
Northridge
0.3
0.2
0.1
0
Loma Prieta
0.3
0.2
0.1
0
Izmit
0.3
0.2
0.1
0
Frankel
(b)
1 2 3 4 5 6
1 2 3 4 5 6
Response Spectra
1 2 3 4 5 6
Fourier Spectra
Time Domain
0.3
0.2
0.1
0
Hartzell
0.3
0.2
0.1
0 0.3
Liu
0.2
0.1
0
Zeng
0.3
0.2
0.1
0
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
Figure 7. Average biases (observed/synthetic) calculated by taking the average of the absolute values of the biases in Figures 4a, 5a, and 6a over (a) the four different methodologies and (b) the three different earthquakes.
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion Loma Prieta
Northridge
1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0.1
1.0
1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0.1
10
Fourier Spectra Standard Deviation
1.0
10
Response Spectra Standard Deviation
Model Avg./Syn (nl log units)
1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0.1
(b)
Response Spectra Standard Deviation
Model Avg./Syn (nl log units)
Model Avg./Syn (nl log units)
Model Avg./Syn (nl log units)
(a)
2447
Fourier Spectra Standard Deviation
1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0
0.1
Frequency (Hz)
10
1.0
1.0 Frequency (Hz)
10
Izmit Model Avg./Syn (nl log units)
(c)
1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0.1
Model Avg./Syn (nl log units)
0
1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0.1
Response Spectra Standard Deviation
(d)
Time Domain Parameter Standard Deviation Northridge
Izmit
0.45
0.45
0.4
0.4
0.4
0.35
0.35
0.35
0.45
0.3
0.3
0.3
0.25
0.25
0.25
0.2
0.2 1
1.0
Loma Prieta
2
3
4
5
0.2 1
2
3
4
5
1
2
3
4
5
10
Fourier Spectra Standard Deviation
1.0
10
Frequency (Hz)
Figure 8. Model epistemic uncertainty measured as the standard deviation in natural-log units of the response spectra and Fourier spectra for (a) Northridge earthquake, (b) Loma Prieta earthquake, (c) Izmit earthquake. (d) Model epistemic uncertainty of the time-domain parameters (1) peak velocity, (2) peak acceleration, (3) cumulative squared velocity, (4) velocity duration, and (5) acceleration duration. Values are standard deviations in natural-log units. of the combined model uncertainty and random variability obtained in the previous section. One notable exception of high values occurs at long periods (7–10 s) for the Izmit earthquake. This rupture is the longest of the three and brings out differences in how directivity and the resulting long-
period velocity pulses are calculated between the different codes. A further inspection of the curves in Figure 8a–c shows that there is greater variability in model predictions at frequencies below 1 Hz than above 1 Hz. This effect can most likely be traced to longer wavelength differences
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Figure 9. a) Estimates of parameter sensitivity expressed as the percentage change in the ground-motion parameter (Fourier amplitude at 0.3, 1.0, and 3.0 Hz and peak velocity) for a change in average rupture velocity from 2.5 to 2:9 km=s for the Northridge earthquake. Two ground-motion simulation methods are considered (Hartzell et al., 2005 and Liu et al., 2006). (b) Same as (a) for the Izmit earthquake. (Continued)
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion
Figure 9. in the slip distributions and the timing of rupture, resulting in different long-period amplitudes, and as previously mentioned, differences in directivity. The corresponding timedomain values of standard deviation are shown in Figure 8d. Again, the standard deviations are approximately one-half the values estimated for the combined model uncertainty and random variability.
Parameter Uncertainty We do not attempt to do a complete study of parameter uncertainty, which for four different codes and the number of parameters required by each code, is beyond the scope of this study. In addition, as has been pointed out by Strasser et al. (2009), a complete study of parametric uncertainty requires the use of joint parameter probability distributions, that is, combinations of parameters that are physically reasonable. Many previous studies have not considered this point and have used randomly determined combinations of parameters,
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Continued.
which can lead to larger values of sigma. To avoid this problem, we constrain our discussion to one important parameter, average rupture velocity, which has a consistent meaning in each code. The fault geometries for the Northridge and Izmit earthquakes are used as our test cases, representing thrust and strike-slip faulting, respectively. Ground motion is calculated on a uniform grid of 100 sites centered above the fault in question, using the Hartzell et al. (2005) and Liu et al. (2006) codes. Figure 1 shows the surface projection of the fault surfaces. Figure 9a,b shows the percentage change in ground motion in going from an average rupture velocity of 2:5 km=s to 2:9 km=s for the Northridge and Izmit earthquakes, respectively. Four measures of ground motion are considered, the Fourier amplitude at 0.3, 1.0, and 3.0 Hz, and the peak horizontal velocity. The patterns are complex; however, certain trends emerge. At lower frequencies, approximately below 1 Hz, directivity effects, causing higher ground motions, are clearly seen up-dip for the Northridge
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S. Hartzell, A. Frankel, P. Liu, Y. Zeng, and S. Rahman
fault and off the ends of the Izmit rupture. However, the pattern of higher ground motion is complicated by the details of the slip distribution. It is postulated that if many different slip distributions were run, the localized highs in ground motion would blend together into a simpler pattern. The pattern of ground motion for the difference in peak velocity generally follows that of the longer-period Fourier components. At higher frequencies, the pattern of the difference in ground motion appears more random, but may still be influenced by the slip distribution. The amplitude of the change in ground motion with rupture velocity increases with decreasing frequency of the ground-motion measure. At 3 Hz, the ground motion increases by a maximum of about 60%; however, at 0.3 Hz, there is an increase of from 150%–300%, depending on the earthquake. The long-period levels are a factor of 2 greater for the Izmit earthquake than for the Northridge earthquake. These observations can be explained by an increase in directivity effects at longer periods and over longer fault lengths caused by a more coherent rupture at these periods and rupture lengths. Previous estimates of parametric uncertainty in ground motion range from about 0.35 to 1.1 natural-log units. Our uncertainty from varying just rupture velocity is largely within this range; however, there are other important source parameters such as slip distribution and rise time, which we have not considered. There are also questions of what are allowable combinations of slip, rupture velocity, and rise time in a kinematic description of faulting. Although studies have attempted to address these questions (Pavic et al., 2000; Guatteri et al., 2003; 2004; Liu et al., 2006; Hutchings et al., 2007; Sorensen et al., 2007; Causse et al., 2008; Ripperger et al., 2008; Schmedes et al., 2010; among others), our knowledge of earthquake ruptures remains limited. Clearly, for Figure 9, there are also strong spatial variations in ground motion with changing source parameters that must be considered. Some areas may be relatively insensitive to a change in source parameters, while others experience significant change in ground motion (Olsen and Mayhew, 2010). Simple examples of source effects responsible for these differences include near-fault directivity and near-fault sensitivity to slip.
Conclusions We have shown that the combined model uncertainty and random variability in ground-motion simulations, based on the results of four independent methodologies and three earthquakes, is in the same range as the values of sigma for regional empirical ground-motion data sets. The standard deviations lie between 0.5 and 0.7 natural-log units for response spectra and 0.5 and 0.8 for Fourier spectra. For the model epistemic uncertainty, most values of standard deviation lie between 0.2 and 0.4, which is about one-half of the estimates for the standard deviation of the combined model uncertainty and random variability. Parametric uncertainty based on variation of just the average rupture velocity is shown to be consistent in amplitude with previous estimates
with percentage changes in ground motion from 50%–300% when rupture velocity changes from 2.5 to 2:9 km=s. In addition, there is some evidence that mean biases can be reduced by averaging ground-motion estimates from different methods.
Data and Resources The strong-motion records used in this study are available from the Center for Engineering Strong Motion Data, http:// www.strongmotioncenter.org (last accessed April 2011) and the COSMOS Virtual Data Center, http://db.cosmos-eq.org/ scripts/default.plx (last accessed April 2011). Ground-motion simulation codes are available upon request from the respective authors.
Acknowledgments The first author benefited greatly from discussions with Melanie Walling and David Perkins. The manuscript was improved by reviews from Melanie Walling, Nico Luco, Arben Pitarka, and an anonymous reviewer.
References Abrahamson, N., and W. Silva (1997). Empirical response spectral attenuation relations for shallow crustal earthquakes, Seismol. Res. Lett. 68, 94–127. Abrahamson, N., and W. Silva (2008). Summary of the Abrahamson and Silva NGA ground-motion relations, Earthq. Spectra 24, 67–97. Abrahamson, N. A., P. G. Somerville, and C. A. Cornell (1990). Uncertainty in numerical strong motion predictions, in Proc. of the Fourth U.S. Natl. Conf. on Earthq. Eng. 1, 407–416. Aki, K., and B. Chouet (1975). Origin of coda waves: Source, attenuation, and scattering effects, J. Geophys. Res. 80, 3322–3342. Al Atik, L., N. Abrahamson, J. Bommer, F. Scherbaum, F. Cotton, and N. Kuehn (2010). The variability of ground-motion prediction models and its components, Seismol, Res. Lett. 81, 794–801. Ameri, G., F. Gallovic, F. Pacor, and A. Emolo (2009). Uncertainties in strong-motion prediction with finite-fault synthetic seismograms: An application to the 1984 M 5.7 Gubbio, central Italy, earthquake, Bull. Seismol. Soc. Am. 99, 647–663. Anderson, J. (2004). Quantitative measure of the goodness-of-fit of synthetic seismograms, in Proc. of Thirteenth World Conf. on Earthq. Eng., Vancouver, British Columbia, Canada, 1–6 August, Pap. No. 243. Anderson, J., and J. Brune (1999). Probabilistic seismic hazard assessment without the Ergodic assumption, Seismol. Res. Lett. 70, 19–28. Anderson, J., J. Brune, R. Anooshehpoor, and S. Ni (2000). New ground motion data and concepts in seismic hazard analysis, Curr. Sci. 79, 1278–1290. Andrews, D. J. (1980). A stochastic fault model. 1: Static case, J. Geophys. Res. 85, 3867–3877. Andrews, D. J. (1981). A stochastic fault model. 2: Time-dependent case, J. Geophys. Res. 86, 10821–10834. Archuleta, R., J., P. Liu, J. H. Steidl, L. F. Bonilla, D. Lavallee, and F. Heuze (2003). Finite-fault site-specific acceleration time histories that include nonlinear soil response, Phys. Earth Planet. In. 4190, 1–29. Atkinson, G. M. (2006). Single-station sigma, Bull. Seismol. Soc. Am. 96, 446–455. Baker, J. W., and N. Jayaram (2008). Correlation of spectral acceleration values from NGA ground motion models, Earthq. Spectra, 24, no. 1, 299–317. Bazzurro, P., B. Sjoberg, and N. Luco (2004). Post-elastic response of structures to synthetic ground motions, Rept. for Pacific Earthquake
Model and Parametric Uncertainty in Source-Based Kinematic Models of Earthquake Ground Motion Engineering Research Center (PEER) Lifeline Program Project 1G00 Addenda, Berkeley, California. Beauval, C., L. Honore, and F. Courboulex (2009). Ground-motion variability and implementation of a probabilistic-deterministic hazard method, Bull. Seimsol. Soc. Am. 99, 2992–3002. Beresnev, I. A., and G. M. Atkinson (1997). Modeling finite-fault radiation from the ωn spectrum, Bull. Seismol. Soc. Am. 87, 67–84. Bommer, J. J., and N. A. Abrahamson (2006). Why do modern probabilistic seismic-hazard analyses often lead to increased hazard estimates?, Bull. Seismol. Soc. Am. 96, 1967–1977. Bonilla, L. F. (2000). Computation of linear and nonlinear site response for near field ground motion, Ph.D. Thesis, University of California, Santa Barbara, 285 pp. Bonilla, L. F., D. Lavallee, and R. J. Archuleta (1998). Nonlinear site response: Laboratory modeling as a constraint for modeling accelerograms, in Proc. of the Second Intl. Symp. on the Effects of Surface Geol. on Seismol. Motion, K. Irikura, K. Kudo, H. Okada, and T. Sasatani (Editors), A. A. Balkema, Rotterdam, 793–800. Boore, D. M. (1983). Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra, Bull. Seismol. Soc. Am. 73, 1865–1894. Boore, D., and G. Atkinson (2008). Ground-motion prediction equations for the average horizontal component of PGA, PGV, and 5%-damped PSA at spectral periods between 0.01 s and 10.0 s, Earthq. Spectra 24, 99–138. Boore, D., W. Joyner, and T. Fumal (1997). Equations for estimating horizontal response spectra and peak acceleration from western North American earthquakes: A summary of recent work, Seismol. Res. Lett. 68, 128–153. Brune, J. N. (1970). Tectonic stress and the spectra of seismic shear waves from earthquakes, J. Geophys. Res. 75, 4997–5009. Brune, J. N. (1971). Correction to tectonic stress and the spectra of seismic shear waves from earthquakes, J. Geophys. Res. 76, no. 20, 5002. Cario, M. C., and B. L. Nelson (1997). Modeling and generation of random vectors with arbitrary distributions and correlation matrix, Tech. Rept. Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois. Causse, M., F. Cotton, C. Cornou, and P.-Y. Bard (2008). Calibrating median and uncertainty estimates for a practical use of empirical Green’s functions technique, Bull. Seismol. Soc. Am. 98, 344–353. Day, S. M., R. Graves, J. Bielak, D. Dreger, S. Larsen, K. B. Olsen, A. Pitarka, and L. Ramirez-Guzman (2008). Model for basin effects on long-period response spectra in southern California, Earthq. Spectra 24, 257–277. Finn, W. D.L., K. W. Lee, and G. R. Martin (1975). Stress-strain relations for sand in simple shear, Soil Mechanics Series No. 26, Department of Civil Engineering, University of British Columbia, Vancouver, British Columbia. Frankel, A. (1991). High-frequency spectral falloff of earthquakes, fractal dimension of complex rupture, b value, and the scaling of strength on faults, J. Geophys. Res. 96, 6291–6302. Frankel, A. (1995). Simulating strong motions of large earthquakes using recordings of small earthquakes: The Loma Prieta mainshock as a test case, Bull. Seismol. Soc. Am. 85, 1144–1160. Frankel, A. (2009). A constant stress-drop model for producing broadband synthetic seismograms: Comparison with the next generation attenuation relations, Bull. Seismol. Soc. Am. 99, 664–680. Graves, R., and A. Pitarka (2010). Broadband ground-motion simulation using a hybrid approach, Bull. Seismol. Soc. Am.100, 2095–2123. Guatteri, M., P. M. Mai, G. C. Beroza, and J. Boatwright (2003). Strong ground-motion prediction from stochastic-dynamic source models, Bull. Seismol. Soc. Am. 93, 301–313. Guatteri, M., P. M. Mai, and G. C. Beroza (2004). A pseudo-dynamic approximation to dynamic rupture models for strong ground motion prediction, Bull. Seismol. Soc. Am. 94, 2051–2063.
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Electric Power Research Institute (EPRI) (1993). Guidelines for determining design basis ground motions, Electric Power Research Institute Tech. Rept. EPRI TR-102293. Hartzell, S., and T. Heaton (1995). San Andreas deterministic fault problem: Green’s function summation for a finite source, in Proc. Modeling Earthq. Ground Motion at Close Distances, J. F. Schneider and P. G. Somerville (Editors), Electric Power Research Institute, TR-104975. Hartzell, S., L. F. Bonilla, and R. A. Williams (2004). Prediction of nonlinear soil effects, Bull. Seismol. Soc. Am. 94, 1609–1629. Hartzell, S., S. Harmsen, A. Frankel, and S. Larsen (1999). Calculation of broadband time histories of ground motion: comparison of methods and validation using strong ground motion from the 1994 Northridge earthquake, Bull. Seismol. Soc. Am. 89, 1484–1504. Hartzell, S., M. Guatteri, P. M. Mai, P. Liu, and M. Fisk (2005). Calculation of broadband time histories of ground motion II: Kinematic and dynamic modeling using theoretical Green’s functions and comparison with the 1994 Northridge earthquake, Bull. Seismol. Soc. Am. 95, 614–645. Herrero, A., and P. Bernard (1994). A kinematic self-similar rupture process for earthquakes, Bull. Seismol. Soc. Am. 84, 1216–1228. Hisada, Y. (2000). A theoretical omega-square model considering the spatial variation in slip and rupture velocity, Bull. Seismol. Soc. Am. 90, 387–400. Hisada, Y. (2001). A theoretical omega-square model considering the spatial variation in slip and rupture velocity II: Case for a two-dimensional source model, Bull. Seismol. Soc. Am. 91, 651–666. Hutchings, L. (1994). Kinematic earthquake models and synthesized ground motion using empirical Green’s functions, Bull. Seismol. Soc. Am. 84, 1028–1050. Hutchings, , L. E. Ioannidou, W. Foxall, N. Voulgaris, J. Savy, I. Kalogeras, L Scognamiglio, and G. Stavrakakis (2007). A physically based strong ground-motion prediction methodology: Application to PSHA and the 1999 Mw 6:0 Athens earthquake, Geophys. J. Int. 168, 659–680. Irikura, K., and K. Kamae (1994). Estimation of strong ground motion in broad-frequency band based on a seismic source scaling model and an empirical Green’s function technique, Ann. Geofis. 37, 1721–1743. Keilis Borok, V. (1959). On estimation of the displacement in an earthquake source and of source dimensions, Ann. Geofis. (Rome) 12, 205–214. Konder, R. L., and J. S. Zelasko (1963). A hyperbolic stress-strain formulation for sands, in Proc. of the Second Pan Am. Conf. on Soil Mech. and Found. Eng., 289–324. Kristekova, M., J. Kristek, P. Moczo, and S. M. Day (2006). Misfit criteria for quantitative comparison of seismograms, Bull. Seismol. Soc. Am. 96, 1836–1850. Lee, M. K.W., and W. D. L. Finn (1978). DESRA-2, dynamic effective stress response analysis of soil deposits with energy transmitting boundary including assessment of liquefaction potential, Soil Mechanics Series No. 38, Dept. of Civil Engineering, University of British Columbia, Vancouver, British Columbia. Lin, P.-S., S.-J. B. Chiou, N. A. Abrahamson, M. Walling, C.-T. Lee, and C.-T. Cheng (2011). Repeatable source, site, and path effects on the standard deviation for empirical ground-motion prediction models, Bull. Seismol. Soc. Am. 101, no. 5, 2281–2295. Liu, P., R. J. Archuleta, and S. Hartzell (2006). Prediction of broadband ground-motion time histories: Hybrid low/high frequency method with correlated random source parameters, Bull. Seismol. Soc. Am. 96, 2118–2130. Ma, S., S. Custodio, R. J. Archuleta, and P. Liu (2008). Dynamic modeling of the 2004 Mw 6.0 Parkfield, California, earthquake, J. Geophys. Res. 113, B02301. Mai, P. M., and G. C. Beroza (2002). A spatial random field model to characterize complexity in earthquake slip, J. Geophys. Res. 107, 1–21. Mai, P. M., W. Imperatori, and K. B. Olsen (2010). Hybrid broadband ground-motion simulations: Combining long-period deterministic synthetics with high-frequency multiple S-to-S backscattering, Bull. Seismol. Soc. Am. 100, 2124–2142.
2452 Morikawa, N., T. Kanno, A. Narita, H. Fujiwara, T. Okumura, Y. Fukushima, and A. Guerpinar (2008). Strong motion uncertainty determined from observed records by dense network in Japan, J. Seismol. 12, 529–546. National Earthquake Hazards Reduction Program (NEHRP) (1994). Recommended provisions for seismic regulations for new buildings, Federal Emergency Management Agency Rept. FEMA 222A, Washington, D.C., 290 pp. Olsen, K. B., and J. E. Mayhew (2010). Goodness-of-fit criteria for broadband synthetic seismograms, with application to the 2008 Mw 5.4 Chino Hills, California, earthquake, Seismol. Res. Lett. 81, 715–723. Pavic, R., M. G. Koller, P.-Y. Bard, and C. Lacave-Lachet (2000). Ground motion prediction with the empirical Green’s function technique: An assessment of uncertainties and confidence level, J. Seismol. 4, 59–77. Pitarka, A., P. Somerville, Y. Fukushima, T. Uetake, and K. Irikura (2000). Simulation of near-fault strong-ground motion using hybrid Green’s functions, Bull. Seismol. Soc. Am. 90, 566–586. Rezaeian, S., and A. Der Kiureghian (2008). A stochastic ground motion model with separable temporal and spectral nonstationarities, Earthq. Eng. Struct. Dynam. 37, 1565–1584. Rezaeian, S., and A. Der Kiureghian (2010). Simulation of synthetic ground motions for specified earthquake and site characteristics, Earthq. Eng. Struct. Dynam. 39, 1155–1180. 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. Schmedes, J., R. J. Archuleta, and D. Lavallee (2010). Correlation of earthquake source parameters inferred from dynamic rupture simulations, J. Geophy. Res. 115, B03304. Schneider, J. F., W. J. Silva, and C. Stark (1993). Ground motion model for the 1989 M 6.9 Loma Prieta earthquake including effects of source, path, and site, Earthq. Spectra 9, 251–287. Silva, W. J., I. G. Wong, and R. B. Darragh (1998). Engineering characterization of earthquake strong ground motions in the Pacific northwest, in Assessing Earthquake Hazards and Reducing Risk in the Pacific Northwest, A. M. Rogers, T. J. Walsh, W. J. Kochelman, and G. R. Priest (Editors), U.S. Geol. Surv. Prof. Pap. 1560, 313–324. Sorensen, M. B., N. Pulido, and K. Atakan (2007). Sensitivity of groundmotion simulations to earthquake source parameters: A case study for Istanbul, Turkey, Bull. Seismol. Soc. Am. 97, 881–900. Strasser, F. O., N. A. Abrahamson, and J. J. Bommer (2009). Sigma: Issues, insights, and challenges, Seismol. Res. Lett. 80, 40–56. Vucetic, M., and R. Dobry (1991). Effects of soil plasticity on cyclic response, J. Geotech. Eng., ASCE 117, 89–107. Walling, M. A. (2009). Non-Ergodic probabilistic seismic hazard analysis and spatial simulation of variation in ground motion, Ph.D. Thesis, University of California, Berkeley. Wang, H., H. Igel, F. Gallovic, A. Cochard, and M. Ewald (2008). Source-related variations of ground motions in 3-D media: Application
S. Hartzell, A. Frankel, P. Liu, Y. Zeng, and S. Rahman to the Newport–Inglewood fault, Los Angeles basin, Geophys. J. Int. 175, 202–214. Zeng, Y. (2006). Broadband strong ground motion simulation using a composite source model, Proc. of Eighth U.S. Natl. Conf. on Earthq. Eng., Pap. No. 8NCEE-001445. Zeng, Y., J. Anderson, and G. Yu (1994). A composite source model for computing realistic synthetic strong ground motions, Geophys. Res. Lett. 21, 725–728. Zeng, Y., J. G. Anderson, and F. Su (1995). Subevent rake and random scattering effects in realistic strong ground motion simulation, Geophys. Res. Lett. 22, 17–20. Zhu, L., and L. A. Rivera (2002). A note on the dynamic and static displacements from a point source in multilayered media, Geophys. J. Int. 148, 619–627.
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