We have continued to develop a multi-disciplinary approach to the problem of predicting of ground motions from scenario earthquakes. Here we present the most recent results on stochastic waveform simulations. Probability Distribution of Strong Ground Motion. Although high-frequency radiation is a major factor in the destructive nature of large earthquakes, the statistical properties of observed accelerograms have received insufficient attention from the seismological community. Studies of the probability distributions of amplitudes of strong ground motions initiated in (Bolt and Abrahamson, 1982; Schenk, 1985) were continued only recently in (Gusev, 1996; Tumarkin and Archuleta, 1997).
Probability distribution curves (i.e., plots of amplitude vs. duration) provide a useful tool for comparing ground motion observations and predictions. Fourier amplitude spectra are widely used for that purpose in the frequency domain. Until now there was no established analog of such comparisons in the time domain. A joint application of these methods will provide an answer to the following important practical problem: how to quantify a similarity between two seismic records.
We developed an efficient approach to calculating probability distributions of amplitudes of seismic signals (Tumarkin and Archuleta, 1984, 1987). If we simply sort the ordinates of a time series in an ascending order, then each value A of the ordinate (observed amplitude) of the resulting function corresponds to the value of the abscissa D, which represents the total duration of shaking with ordinates less than A. After inverting this monotone function and normalizing by the total length of the initial time series we obtain the probability distribution function.
In Figure 1 we show the application of this procedure to the East component of particle acceleration recorded at the Santa Susana site (DOE-USGS) during the Northridge mainshock. First we take acceleration time series, sort the ordinates and then interchange the abscissas and ordinates (top row). The resulting function after normalizing by the total length of time series (30 s in this case) represents the probability distribution of the accelerogram. In the bottom row we show the same procedure applied to absolute values of the ground acceleration which leads to a one-sided distribution function.
In Figure 2 we compare the probability distribution of acceleration obtained in Figure 1 to the Gaussian and Cauchy distributions. The solid line representing the observed distribution has a very different shape than the Gaussian curve (dotted line). At the same time, the Cauchy distribution (dashed) is much more similar to the observed one. That fact is especially well illustrated by the right panel of Figure 2, showing the corresponding probability density functions. Although this is only one accelerogram, it clearly shows that the Gaussian distribution does not adequately represent the larger amplitudes of ground motion - the levels that are most responsible for damage.
The fact that accelerograms are characterized by "a heavy-tailed amplitude distribution, with the enhanced probability of large peaks" was also noted by A. Gusev. A very important practical conclusion reached in (Gusev, 1996) is that "in calculations for the stochastic strong motion prediction, the use of peak factor estimates based on the stationary Gaussian model may underestimate peak acceleration value, by something like 25% on the average. Much larger underestimation can be expected if this model is used to estimate a rare event, such as peak acceleration with the repeat time of, say, 1000 years" That point is of vital importance to the seismic hazard assessment in the central and eastern United States (CEUS). Indeed the CEUS seismicity is characterized by infrequent large earthquakes. Due to an insufficient number of strong ground motion recordings, the attenuation models forming the basis of the probabilistic seismic hazard analysis in the CEUS are based on a stochastic approach for generating ground motions (e.g., Schneider, 1993; Frankel et al., 1996). Modified stochastic approach. The basic idea of the stochastic approach (SA) is that the ground motion is represented by a windowed and filtered white noise time series, where the average spectral content and the duration over which the motion lasts are determined by a seismological description of seismic radiation that depends on source size (Boore, 1983).
Recently we found ways to improve upon the standard stochastic ground motion modeling techniques to account for observed non-Gaussian probability distribution of strong motion records. Here are our modifications:
* Based on the observations that the amplitude distribution of acceleration amplitudes is non-Gaussian we use the uniform distribution for the random number generation without transforming it into the normal distribution.
* In order to account for directivity we adjust the apparent source corner frequency according to the source-receiver geometry.
* We apply an envelope function to account for both S-P times (calculated from the hypocentral distance), differences in S and P amplitudes and attenuation. The shape of the envelope is chosen to be the BruneUs pulse: t*exp(-pt/T), where t is time, T= T(f0,R) is the duration.
* To account for the high-frequency attenuation (Anderson and Hough, 1984) and the impedance contrast I between the source and the site, we are using the following functional form of the frequency filter that describes the attenuation and site amplification effects: I*cosh(p*k*f)/(cosh(2*p*k*f)+I-1).
This procedure produces realistically looking time series. In Figure 3 we show the comparisons of the acceleration (left column) and velocity (right column) time histories at the Santa Susana site with ten stochastic realizations. Three components of observed records are plotted at the top. The predicted waveforms compare well with observations both in amplitude and duration. The work on this project resulted in the following publications:
Papers:
Archuleta, R. J., Liu, P.-C. and A. G. Tumarkin (1997). Source Inversion and Ground Motion Prediction with Empirical Green's Functions. In "Proceedings of the Northridge Earthquake Research Conference" CUREe, Berkeley, CA.
Tumarkin, A. G. and R. J. Archuleta (1997). Recent Advances in Prediction and Processing of Strong Ground Motions, Natural Hazards, 15, 199-215.
Abstract:
Tumarkin, A. G. and R. J. Archuleta (1997). Stochastic Ground Motion Modeling Revisited: Seismological Research Letters, 68, No. 2, p. 312.
