Hazards Evaluation Program
Projects GT-1 & GT-9



Ground Motion Amplification of Soils in the Upper Mississippi Embayment


Stochastic Approach for Modeling Rock Motions


Rock motions in the Central United States were modeled using a stochastic approach that assumes seismic shear wave energy may be represented as band-limited white noise (BLWN) and peak parameters are determined from random vibration theory (RVT) (Hanks and McGuire, 1981; Boore, 1983).   The stochastic approach assumes that an acceleration time history may be approximated by band-limited, white Gaussian noise with a Fourier amplitude spectrum given as

where S(f,M0) is the source spectrum, D(f,R) is a diminution factor accounting for attenuation, and P(f) models site effects that filter high-frequency energy.   The source spectrum is based on the w2 (Brune) point source model (Brune, 1970, 1971).   The diminution factor includes both anelastic and geometric attenuation.   The low-pass filter, P(f), is a function of the spectral decay parameter, k(0), and accounts for filtering of high-frequency seismic shear energy (Anderson and Hough, 1984).

The diminution factor, D, selected for this study is based on the model developed by Atkinson and Boore (1995) for Eastern North America.   The values of the spectral decay parameter, k(0), are based on Herrmann and Akinci (2000) for the Mississippi Embayment and Frankel et al. (1996) for rock sites in the Central U.S.   The k(0) value for the Mississippi Embayment is 0.048 sec compared with 0.006 sec for hard rock sites.

Figure 1 shows a sample Fourier amplitude spectrum for a point-source model with one-corner frequency and a k(0) of 0.048 sec, a moment magnitude, Mw, of 6.5, at an epicentral distance of 50 km.   Rock motions were generated for moment magnitudes of 5.5, 6.5, 7.5, and 8.0 and epicentral distances of 10, 25, 50, 100, and 200 km.   The stochastic approach was used to generate rock motions at the base of the embayment.   A one-dimensional, equivalent-linear analysis was used to propagate the rock motions through the soil column and calculate ground motions at the surface.





Figure 1 Sample Fourier Amplitude Spectrum


Site response analyses were conducted entirely in the frequency domain.   Peak parameters such as peak ground acceleration (PGA) and peak strains were calculated using random vibration theory (RVT).   Random vibration theory relates peak parameters in the time domain to root-mean-square (rms) parameters (Cartwright and Longuet-Higgins, 1956).   The rms parameters are calculated using Parseval's Theorem which relates the total energy in the frequency domain to the total energy in the time domain (Vanmarcke and Lai, 1980).


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References

  • Anderson, J.G. and S.E. Hough (1984), "A Model for the Shape of the Fourier Amplitude Spectrum of Acceleration at High Frequencies," Bulletin of the Seismological Society of America, Vol. 74, No. 5, pp. 1969-1993.
  • Atkinson, G.M. and D.M. Boore (1995), "Ground-Motion Relations for Eastern North America," Bulletin of the Seismological Society of America, Vol. 85, No. 1, pp. 17-30.
  • Boore, D.M. (1983), "Stochastic Simulation of High-Frequency Ground Motions Based on Seismological Models of the Radiated Spectra," Bulletin of the Seismological Society of America, Vol. 73, No. 6, pp. 1865-1894.
  • Brune, J.N. (1970), "Tectonic Stress and Spectra of Seismic Shear Waves from Earthquakes," Journal of Geophysical Research, Vol. 75, No. 26, pp. 4997-5009.
  • Brune, J.N. (1971), "Correction," Journal of Geophysical Research, Vol. 76, p. 5002.
  • Cartwright, D.E. and M.S. Longuet-Higgins (1956), "The Statistical Distribution of the Maxima of a Random Function," Proc. of the Royal Society of London, Series A (Mathematics and Physical Science), Vol. 237, pp. 212-232.
  • Frankel, A.D., C. Mueller, T. Barnhard, D. Perkins, E.V. Leyendecker, N. Dickman, S. Hanson, and M. Hopper (1996), National Seismic-Hazard Maps: Documentation, USGS Open-File Report 96-532.
  • Hanks, T.C. and R.K. McGuire (1981), "The Character of High-Frequency Strong Ground Motion," Bulletin of the Seismological Society of America, Vol. 71, No. 6, pp. 2071-2095.
  • Vanmarcke, E.H. and S.-S.P. Lai (1980), "Strong-Motion Duration and RMS Amplitude of Earthquake Records," Bulletin of the Seismological Society of America, Vol. 70, No. 4, pp. 1293-1307.

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Contents

  • Stochastic Approach for Modeling Rock Motions



Updated by S. Romero,   May 23, 2001