by Gail M. Atkinson and Paul G. Somerville. Abstract Ground-motion time histories for use in engineering analyses of structures in eastern North America are ...
Bulletin of the Seismological Society of America, Vol. 84, No. 2, pp. 400-414, April 1994
Calibration of Time History Simulation Methods by Gail M. Atkinson and Paul G. Somerville
Abstract
Ground-motion time histories for use in engineering analyses of structures in eastern North America are often simulated from seismological models, owing to the paucity of real recordings in the magnitude and distance ranges of interest. Two simulation methods have been widely used in recent years: the stochastic method and the ray-theory method. In the stochastic method, as implemented in this study, ground motion is treated as filtered Gaussian noise whose underlying spectrum is determined from an empirical region-specific seismological model of the source and propagation processes. In the ray-theory method, as implemented in this study, the ground motions are simulated by convolving an empirical source function with theoretical Green's functions for a specified crustal structure model. This article compares results of the two simulation methods for four well-recorded "calibration" events and assesses the applicability of the methods. The assessment is based on comparisons of groundmotion parameters from the simulated data with those of the actual recordings. Ground-motion parameters in the frequency range from 1 to 10 Hz are satisfactorily predicted by both methods. Averaged over the four events studied, the stochastic method underpredicts 1-Hz response spectra by 20 to 40% but accurately predicts response spectra for frequencies of greater than 2 Hz; it also accurately predicts peak ground acceleration and velocity. The wave-propagation method underpredicts 1-Hz response spectra by 10 to 40% but accurately predicts response spectra for higher frequencies; it overpredicts peak ground acceleration and velocity by 10 to 40%. Both methods are imprecise: the standard error of an estimate is a factor of about 2.2. The bias and standard error of an estimate for the wave-propagation method are generally slightly lower than for the stochastic method, if the focal depth of the event can be specified (i.e., as for a past earthquake). If the focal depth of the event is not known (i.e., as for a future earthquake) then the accuracy and precision of t h e t w o methods are about the same. The chief advantage of the wave-propagation method is its predictive power; since its attenuation function is derived from the focal depth and crustal structure it does not require knowledge of the empirical attenuation function. The chief advantage of the stochastic model is its economy and simplicity.
Introduction late the required records, based on seismological models of earthquake ground-motion generation processes. Two simulation methods have been widely used in recent years: the stochastic method and the ray-theory method. In the stochastic method, ground motion is treated as bandlimited, finite-duration Gaussian noise, whose underlying spectrum is determined from a seismological model of source and propagation processes. In the raytheory method, the ground motions are simulated by convolving an empirical source function, usually derived from a past event, with theoretical Green's functions for
Seismic-hazard studies are often used to identify "design earthquakes," in terms of magnitude and distance from a site of interest. Ground-motion time histories for the desired magnitude(s) and distance(s) are then sought, to be used as input for engineering analyses of a proposed structure. For regions such as California, where strong ground-motion records are relatively abundant, a suite of recordings from past earthquakes is usually available. In eastern North America, by contrast, relevant records for the desired magnitudes and distances are seldom available. It is therefore necessary to simu400
Calibration of Time History Simulation Methods
401 controlled circumstances in which: (1) the source modons are specified; and (2) there are recorded data against which the simulations can be compared. In these circumstances we can make a direct assessment of the ability of the two methods, as implemented here, to accurately model propagation effects. The methods are used to simulate ground-motion recordings for four events which were recorded on the Eastern Canada Telemetred Network (ECTN) (see Table 1 and Fig. 1), and these simulated data were compared with the actual records. For these comparisons, the se-
a specified crustal structure model. In essence, the raytheory approach takes a more deterministic view toward ground-motion prediction than does the stochastic method. A third method, developed by Ou and Herrmann (1990), can be considered a hybrid that is intermediate to the other two methods, in that it uses ray theory to predict wave-propagation effects, but treats the phasing of individual rays as random. The Ou and Herrmann (1990) method is not investigated in the current study. The purpose of this study is to compare results of the stochastic and ray-theory simulation methods under
Table 1 Source Parameters of Study Events* Date (m/d/yr)
Location
10/11/83 11/25/88 03/03/90 10/19/90
Ottawa, Ontario Saguenay,Quebec Charlevoix,Quebec Mont Laurier, Quebec
Coordinates
(45.2, (48.1, (47.9, (46.5,
Depth
mN
M
Atr
14 26 20 7
4.1 6.5 3.6 5.1
3.6 5.8 3.4 4.7
40 520 70 520
75.8) 71.2) 70.0) 75.6)
*The term m~ is the Nuttli magnitude, as quoted by the Geological Survey of Canada. M is moment magnitude. A@is the Brune stress drop in bars, required to match the observed high-frequency spectral amplitudes. Depth is the focal depth in kilometers.
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402
G.M. Atldnson and P. G. Somerville
lected ground-motion parameters were peak ground acceleration (PGA), peak ground velocity (PGV), and response spectral ordinates (pseudo-acceleration, PSA) at frequencies of about 1, 2, 5, and 10 Hz. We begin by reviewing empirical studies that were used to determine the source and site terms applied in both simulation methods, and the attenuation function applied in the stochastic method. The simulation methods are described next. We then discuss the degree-offit to the data achieved by the two methods and present conclusions as to their applicability. Empirical Description of Source, Site, and Attenuation Terms A recent empirical study used approximately 1200 recordings from the ECTN, covering the magnitude range from 3.0 to 6.5 and distances from 10 to 1000 km, to make an empirical determination of source, site, and attenuation parameters for eastern North American (ENA) earthquakes (Atkinson and Mereu, 1992; Atkinson, 1993a). The empirical study fit Fourier spectral amplitude data to an equation of the form:
Calibration Events:
log Aij(f) = log Ei(f) - b log R o
- c(f)Rq + log Sj(f)
empirically-derived attenuation form is used as the basis for describing the decay of spectral amplitudes with distance for the stochastic simulations of this study. The regression analysis also determined the site terms, St(f), which modify ground-motion amplitudes at each station, under the constraint that the average of the site terms over all rock stations be equal to zero. Since all stations except one are located on rock, site terms are generally within a factor of 1.5 of unity, although some exceed a factor of 2. (Site terms are listed in Atkinson and Mereu, 1992.) The source spectra (E,(f)) for each calibration event, also determined by the regressions, are displayed in Figure 2. The source spectra may be fitted to the Brune (1970) model to obtain the seismic moment and stress drop of each event, as given in Table 1. As discussed by Atkinson (1993a), however, the shapes of the spectra do not necessarily agree well with the Brune shape. The use of these regression results to constrain the model parameters of the simulations should result in a better fit of "predictions" to observations than would normally be expected for a future earthquake. In other
(1)
where Ao(f) is the observed amplitude of earthquake i at station j, for frequencyf, Ei(f) is the source amplitude of earthquake i, Sj(f) is the site term for station j, R is hypocentral distance, b is the geometric spreading coefficient, and c(f) is the coefficient of anelastic attenuation (which includes the effects of scattering). Equation (1) was applied to the shear-wave window, including both the direct wave and all significant reflected and refracted S arrivals; these phases contain all of the significant ground motion from an engineering point of view. To accommodate complexity in the shape of the attenuation curve resulting from wave-propagation effects, the b coefficient was allowed to take on different values for different distance ranges, corresponding to the attenuation of different seismic phases. The results showed that the attenuation curve for the Fourier amplitude of acceleration in ENA, for frequencies of 1 to 10 Hz, consists of three segments. Amplitudes decay slightly faster than 1/R (b = 1.1) at (hypocentral) distances from 10 to 70 km, corresponding to attenuation of the direct wave. Between 70 and 130 km, where the direct wave is joined by S,,S (Burger et al., 1987; Somerville et al., 1990), spectral amplitudes are approximately constant. Beyond 130 km, corresponding to the Lg phase, amplitudes decay at a rate that is consistent with R -°5 and Q = 6 7 0 f °'33, where Q = 2.3~rf/ (cfl) (fl is the shear-wave velocity, 3.8 km/sec). This
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408 nation outside the 1- to 10-Hz frequency band. The record is transformed back to the time domain to yield an instrument-corrected accelerogram. 3. The peak acceleration is obtained directly from the accelerogram. Note that this is not a true measure of peak acceleration because frequencies above 12 Hz have been removed. 4. The 5% damped response spectrum is computed from the accelerogram; values of pseudo-acceleration (PSA) are tabulated for frequencies of 1, 2, 5, and 10 Hz. Figures 8 and 9 compare the response spectra of the simulated and real recordings on an event-by-event basis, for frequencies of 1 and 10 Hz, respectively. In judging the comparisons it is important to distinguish between accuracy and precision. The figures indicate that the simulated ground motions are reasonably accurate (i.e., correct on average), but individual estimates are highly imprecise; the real records show large random variability, which we did not attempt to reproduce in the simulations. The Charlevoix earthquake, in particular, shows extreme amplitude variability. The reason for this is unknown; examination of many other ECTN recordings suggests that the variability shown by the Charlevoix event is atypical. Note that the near data (R < 150 km) for the Saguenay event, which consistently exceed the predicted amplitudes, are from strong-motion recordings for which we have no knowledge of the site terms. These data were not used in the regressions of Atkinson and Mereu (1992) that determined the source spectra. These stations may have a significant site response; alternatively their large amplitudes may be the result of unmodeled effects such as directivity (Haddon, 1992). Figures 10 and 11 plot the differences between the recorded and simulated seismograms for several groundmotion parameters, combining all events. The figures plot (log, base 10) residuals as a function of (log) distance, for PGA, PGV, and PSA at 1, 2, 5, and 10 Hz; the log residuals are defined as the difference between the log of the observed record parameter and the log of the simulated record parameter. Thus a log residual of 0.3 for PGA, for example, would indicate that the observed PGA was a factor of 2 (= 10°3) larger than the PGA of the simulated record for that event and station. The residual plots indicate that the simulations from both methods match the observations reasonably well. The stochastic simulations tend to underestimate the ground motions for frequencies of 1 to 2 Hz, particularly in the distance range from 100 to 400 km. At 10 Hz, the stochastic simulations tend to underestimate the ground motions at distances less than 100 km, and overestimate them at larger distances. This could mean that the adopted frequency-independent attenuation form is not matching frequency dependencies in the actual attenuation. In particular, the flattening of the attenuation curve as a result
G.M. Atkinson and P. G. Somerville of the first postcritical reflection from the Moho may be more pronounced at low frequencies than at high frequencies. Alternatively, the long observed durations of high-frequency motions may be masking a shorter duration that applies to the more-coherent low-frequency components of the signal. These discrepancies could be resolved by fine-tuning the attenuation and other parameters. (They could also be reduced by eliminating the Saguenay strong-motion data, which were not used in determining the Saguenay source spectrum because of the unknown site terms.) Since the match between synthetics and recordings is reasonable, such f'me-tuning may not be warranted for practical purposes. Examining the ray-theory simulations in Figures 9 and 10, there are no apparent trends with distance for frequencies of 1 and 2 Hz, indicating that the shape of the attenuation of the data is adequately matched by the wave-propagation method. For higher frequencies and for peak velocity and acceleration, there is a tendency for underprediction between 60 and 100 km, and overprediction between 100 and 160 km. This suggests that the flattening resulting from the critical reflections is slightly less pronounced at high frequencies in the data than in the wave-propagation model. Tables 2 and 3 summarize the differences between the recorded and simulated ground-motion parameters, listing the mean residual (Table 2) and standard error (Table 3) of an estimate for three distance ranges (R < 100 km, R < 200 km, and R < 1000 kin) for each simulation method. The mean residual is a measure of model bias whereas the standard error measures variability of individual estimates. The near-zero mean of the residuals indicates that the predictions are accurate on average, while the large standard errors (corresponding to more than a factor of 2) indicate low precision for any given event and station. The variability of estimates is large considering that this represents only the intra-event variability (i.e., the source spectra are known). The resuits of regression to the larger ECTN dataset of 1200 records, by contrast, indicate that the average standard deviation of data from regression predictions (intra-event variability only) is about a factor of 1.6. The ray-theory method has used an additional parameter not accounted for in the empirical attenuation model, namely the focal depth of the event. In general, this parameter is not known for future events. To measure the performance of the wave-propagation method under the same conditions as used in the stochastic method, we calculated synthetic seismograms using a depth of 11 km (average depth for eastern Canadian earthquakes) for each event, instead of the actual focal depths (which range from 7 to 26 km). The residuals for this case are also shown in Tables 2 and 3. The degreeof-fit of these simulations to the data is not as good as that obtained when the actual depths were used. This indicates that the focal depth does influence the recorded
Calibration of Time History Simulation Methods
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